Why did Bilbo want the ring back from Frodo?

The ring turns its owner selfish and paranoid. It stirs up greed, petty anger, fear and other monstrous emotions. That's the corrupting power of the ring. And that's what surged up in Bilbo during that encounter.

Think of the Ring like an addiction- Frodo is just starting out with it at this point, so it hasn't affected him too much yet even though he can feel it is pretty powerful. Bilbo has had it for at least 50 years now, and he's quite, quite attached to it. It's very hard for him to give it up cold turkey, and he wants to jump back on the bandwagon. Gollum is at the point where he cannot live without it, and it drives everything he does and tries to get at.

As I understood that scene was meant to emphasize the power that the ring could wield over the holder even after just a few years of Bilbo having it in his ownership. The stark contrast showed the audience that Bilbo's' personality itself had not changed THAT much but in matters concerning the ring any former holder would become very overprotective and greedy basically the same personality traits that multiplied a hundredfold lead a person to become Gollum-like(yes I am going to invent that word)

This was not first time that Bilbo has tried to keep Ring with himself. Initially, also Gandalf needed to force him to give Ring. This is effect of ring on peoples that they become attached or attracted towards the Ring. We saw this effect in movie multiple times over different persons like Isildur, Boromir. Even Gandalf was not ready to carry it to Mount Doom as it will corrupt his mind. Frodo also hesitated in the end while reached to Mount Doom due to same effect. Even when he went through to destroy it and knew that it is evil. This effect is more on Ring Bearer or person who keeps the ring with himself. You can see Gollum has dedicated his life for his percious "Ring" and even in end , while dying in Doom he kept his hand up trying to save.

Wikia also confirms these effects :-

For mortals, the ring had several side effects, most of them negative. Perhaps the first was that the bearer of the Ring almost immediately began to develop an unusually strong attachment to it, and would be unwilling to give it up or abandon it.

Firstly you have to examine what the ring is. It is an extension of Sauron's will, a physical item that Sauron imbues with his own spirit through magical means in an effort to not only make him stronger but to extend his dominance over other magical rings that are owned by the leaders of the different races of Middle Earth. The creation of the ring follows a period of time in which Sauron along with the Elven smiths of Eregion, most notably Celebrimbor crafted 19 rings of power for the chief Lords of the children of Iluvatar. Three were given to the Elven Kings. One ring to Gil Galad High King of the Noldor and the ally of Elendil High King of Gondor and Arnor joining forces with the Numenorean during the War of the Last Alliance. After Gil Galad was slain by Sauron in that war the ring was kept by Elrond. Another ring was given to Cirdan the shipwright, the oldest living elf in Middle Earth during the second age and Lord of the Grey Havens. Cirdan was gifted with foresight and willingly gifted his ring into the protection of Gandalf when the Istari arrived in Middle Earth much to the chagrin of Saruman, their leader. The third ring was gifted to Galadriel, the lady of Lorien who continued to hold onto her ring throughout the Third Age.

Sauron also gave seven of these rings to the Dwarf Lords and Nine rings to great Kings of Men. The targets of Sauron's rings all held some degree of influence over the different free peoples and if Sauron could dominate them then he would control all life on Middle Earth. I've highlighted the Elven rings specifically because out of the 19 rings (excluding the numerous practise attempts) that were crafted by the Elven smiths of Eregion, those three were crafted by Celebrimbor alone and were notably free of the influence of Sauron. Celebrimbor was himself a descendant of Feanor, who had crafted the three Silmarils from the light of the two trees before the First Age of Middle Earth, and was considered to be the greatest smith in Middle Earth at the time. Celebrimbor was also, along with Narvi the Dwarf, the creator of the Doors of Durin that sealed the Western Gate of Moria.

At the time these rings were constructed Sauron was still able to take a fair form, the form of Annatar, which he used to deceive the Noldor and was therefore able to use the Eregion smiths, called the Gwaith-I-Mirdain to forge his rings. Celebrimbor however distrusted Sauron which is why he chose to make the Elven rings alone and after their construction, hid them from Sauron. After the 19 rings had been made Sauron had in secret been creating the One Ring with the sole intent of binding the others to him and thus the wearers of these rings would be under his sway and by extension he could dominate the population of Middle Earth.

Because Celebrimbor had secretly crafted the Elven rings alone, when Sauron came to place the One Ring on his finger, the Elves became immediately aware of his presence and removed their rings. This infuriated Sauron who declared open war on the Elves demanding the rings be handed over to him. The elves however hid their three (the greatest of the rings of power) as well as passing one of the rings onto the Dwarves of Khazad Dum who had good relations with the Gwaith-I-Mirdain. Sauron however was able to reclaim the other 15. Of those 15 he gifted 6 to the Dwarves whilst nine he gave to Men knowing they would bend more easily to his will. Of the recipients of the rings the nine Men would eventually be dominated completely by the will of Sauron and became the Nazgul, his chief lieutenants. The seven Dwarf Lords would initially only see great increases in their wealth which over time instilled in them an overwhelming greed for treasure that would in turn lead to their ruin.

As for the three Elven rings, they were not created with the intent of war or dominance but in preservation of the Elven domains and healing and so they enabled their users to protect the Elven kingdoms that remained in Middle Earth. All the rings however were bound to the One Ring and whoever wore the One Ring could see the thoughts of those wearing the three elven rings despite Sauron's influence not being involved in their construction. All of these rings however had their power tied to the One Ring so when the One Ring was destroyed in Mount Doom, the other rings lost their power.

This is important to know in relation to the One Ring because it demonstrates its purpose. It is an object created solely with the intent of dominating all life in Middle Earth. Not only that but it was created by a being that used deceit and manipulation to achieve many goals. To look at the Fall of Numenor for a moment. At the time of its fall, Numenor was ruled by a King known as Ar-Pharazon who was an extremely capable martial leader. Ar-Pharazon was incredibly prideful and when it came to his attention that Sauron, who was dwelling in Mordor at the time, was calling himself the Lord of Men Ar-Pharazon decided to challenge Sauron's power directly. He marched on Mordor with an army so powerful it caused Sauron's forces to desert him and Ar-Pharazon would take Sauron back to Numenor in chains.

Sauron however feigned awe at Ar-Pharazon's might and upon landing in Numenor, worked to gain the King's trust which he accomplished through flattery, lies and half truths. Eventually Ar-Pharazon made Sauron his chief advisor who managed to corrupt the King and turn him to the worship of Morgoth claiming Morgoth could help the King cheat his own mortality. Sauron was able to erect a great temple to Morgoth on the island, persecute the faithful and chop down the white tree of Numenor. As the King grew old Sauron convinced Ar-Pharazon to take immortality by force and convinced him to assault the Undying Lands. The King reached the shores of Valinor with a great fleet and the Valar laid down their guardianship of the world calling upon Iluvatar to intervene. Eru Iluvatar sank Ar-Pharazon's fleet along with the island of Numenor allowing only the faithful Numenorean's under the leadership of Elendil and his sons Isildur and Anarion who sailed to Middle Earth where they founded the kingdoms of Gondor in the south and Arnor in the north.

With Eru's sinking of Numenor and reshaping of the world to separate the Undying Lands from the world (stopping mortals from travelling there), Sauron himself suffered his first "death" as his physical body was destroyed, though his spirit was not diminished and he escaped with the ring back to Mordor where he rebuilt a new body, though he could no longer take the fair form of Annatar. From this you can see how skilful Sauron was with words alone. He was able to manipulate a King that had conquered him and taken him prisoner into trusting him completely to the point that he even assaulted the Valar on Sauron's instruction.

Within "The Lord of the Rings" we know that Sauron poured an extension of his spirit into the One Ring when he crafted it, linking it to him and allowing him to dominate the other rings. The One Ring itself is often described as if it is alive, with a will of its own, desperately seeking to return to its master Sauron. This is a fact that Gandalf informs Frodo of. Aragorn also tells Frodo that the Nazgul will always be drawn to the One Ring because of its connection to them and Sauron. The Ring even manipulates Gollum into at first stealing the Ring from Frodo and then later into taking them to Shelob where the Hobbits will likely die and Gollum will claim the Ring once again. If Gollum is successful he will likely use the Ring again to seek out food as he does in "The Hobbit." As soon as he wears it, the Nazgul will come for him.

Gollum is a being that has been in possession of the Ring for a considerable amount of time prior to the events of "The Hobbit." We see how deeply the Ring has affected him to the point where Gollum has developed an entirely new personality, that of the Ring. In fact this personality had become so dominant that Gollum had completely forgotten his own identity as Smeagol which he only begins to remember again after Frodo brings it up. Bilbo by the time of "The Lord of the Rings" has already had the Ring in his possession for 60 years. His 111th birthday is even explained as being an unusual age for a Hobbit to reach. When Gandalf meets with his old friend again he is surprised to see that Bilbo despite his advanced age still appears quite youthful.

This is the Ring affecting Bilbo physically. Because of the events of Bilbo's birthday, Gandalf begins to suspect that the magic ring is something more than it seems which sees him go on his quest to gather information which leads him to the conclusion that Bilbo's ring is indeed the One Ring. When Frodo meets Bilbo in Rivendell and Bilbo momentarily tries to attack Frodo, this is merely an extension of the impact the ring has had on Bilbo. The Ring after all seeks to dominate all life in the name of its master, when someone other than Sauron uses the Ring, its sole intent is to corrupt that individual into submitting to Sauron as a means of returning to Sauron. The easiest way to corrupt an individual is to appease their desire. For someone like Boromir for example, the Ring preys on Boromir's desire to protect the people of Gondor by influencing his thoughts and making him despair in the belief of what may happen if he doesn't use the Ring to save Gondor. This is why Gandalf refuses to take the Ring and even go as far as to tell Frodo that even if he took the Ring with a desire to do good, through him the Ring would wield a power too terrible and evil to imagine.

In essence the Ring is simply a manifestation of the old adage, "The road to Hell is paved with good intentions." It makes its user desire the power it promises by preying on desire. Therefore Bilbo in his 60 years of possessing the Ring, though perhaps slower than most beings given Hobbits seemingly natural resilience to the Ring, is also corrupted by it. We are informed directly of the odd physical appearance of Bilbo despite his age which is a clear indicator of one of the Rings many abilities, sustaining the bearer granting them an unnatural lifespan. If Tolkien has indeed examined the importance of this detail then it goes without saying that the Ring has also had a mental impact on Bilbo. After all Gollum's mental state is about as twisted as his physical appearance.

Bilbo wanted the Ring back because its corrupting nature had left him desiring it, just as Gollum desired the Ring and Isildur before him. Both for different reasons. Gollum prized it as a beautiful treasure, similarly to Bilbo who also saw it as a useful object. Isildur prized it as a trophy and symbol of his might, having vanquished Sauron in battle. Boromir desired it to help his people, as did Galadriel though she was able to resist in a fashion similar to Gandalf. Saruman desired it as a means of domination himself. Frodo himself succumbs to the Ring's corrupting influence. Other than Tom Bombadil who's nature is a mystery, only Faramir seems to be able to resist the corrupting influence of the Ring which is interesting as the Ring's effect on Men was greater than what it held over other races largely because of the Ring's ability to sustain life unnaturally allowing the Ring to deceive Men into believing they could cheat their mortality. In resisting however Faramir states:

"But fear no more! I would not take this thing, if it lay by the highway. Not were Minas Tirith falling in ruin and I alone could save her, so, using the weapon of the Dark Lord for her good and my glory. No, I do not wish for such triumphs, Frodo son of Drogo."

He is a man of honour and vows to hold true to his word. That's not to say the Ring doesn't try and tempt him in some way but Faramir who sticks to his word refuses to even consider using the weapon of such an enemy for his own devices even when not entirely certain what it is the Hobbits carry. When he does discover that Frodo and Sam are carrying the Ring, he also learns from Sam how the Ring had corrupted his brother and realises the peril of such a thing which allows him to completely resist it. Basically none of the characters in Middle Earth save Sauron and Tom Bombadil could remain completely unaffected by the Ring's power and those that had been around it for longer and benefitted from it in some way, desired it far more greatly than others.

In short: The ring's influence on him (similar to how Gollum wants the Ring back as well). In the hands of Bilbo, the ring is far easier to be reclaimed by Sauron, as such it tries to manipulate him once again.