Why does Bilbo Baggins decide to go with the dwarves in "The Hobbit: An Unexpected Journey"

In the book, he actually goes out of a sense of duty to fulfill a contract which he was tricked into accepting by Gandalf. It's a matter of his good name and integrity. However, the movie changed things slightly by making Bilbo originally refuse to join the journey, but then the next morning he changes his mind.

In hobbit culture, it is considered taboo to seek out adventures and excitement. Hobbits are fans of predictability and routine. This culture has been set in Bilbo his entire life and he tries to be a proper and respectable hobbit. That said, he also enjoys reading stories about elves and far off lands. In the book it is mentioned how he loves Gandalf's fireworks and is very imaginative as a young hobbit.

There is a conflict in the character: be a respectable hobbit and do the easy thing by staying home, or go on an adventure and leave his home behind. This conflict is a recurring theme throughout the movie and the book. When he wakes up in the morning and finds the dwarves gone, at first he is relieved, but as he sits in the quiet and ponders to himself he realizes that he can't pass up the chance to go on an adventure. Hoping he's not too late, he signs the contract and sets out to join the dwarves.

He is coerced into the adventure by Gandalf, who both realises he needs a hobbit and sees the potential for adventure in Bilbo.

In the novels, Bilbo is very against the adventure to begin with. From Chapter 1:

“Sorry! I don’t want any adventures, thank you. Not today. Good morning! But please come to tea—any time you like! Why not tomorrow? Come tomorrow! Good bye!” With that the hobbit turned and scuttled inside his round green door, and shut it as quickly as he dared, not to seem rude. Wizards after all are wizards.

“What on earth did I ask him to tea for!” he said to himself, as he went to the pantry. He had only just had breakfast, but he thought a cake or two and a drink of something would do him good after his fright.

Gandalf in the meantime was still standing outside the door, and laughing long but quietly. After a while he stepped up, and with the spike on his staff scratched a queer sign on the hobbit’s beautiful green front-door. Then he strode away, just about the time when Bilbo was finishing his second cake and beginning to think that he had escaped adventures very well.

Then, from Chapter 2:

“What message?” said poor Mr. Baggins all in a fluster.

“Great Elephants!” said Gandalf, “you are not at all yourself this morning—you have never dusted the mantelpiece!”

“What’s that got to do with it? I have had enough to do with washing up for fourteen!”

“If you had dusted the mantelpiece, you would have found this just under the clock,” said Gandalf, handing Bilbo a note (written, of course, on his own note-paper).

This is what he read:

“Thorin and Company to Burglar Bilbo greeting! For your hospitality our sincerest thanks, and for your offer of professional assistance our grateful acceptance. Terms: cash on delivery, up to and not exceeding one fourteenth of total profits (if any); all travelling expenses guaranteed in any event; funeral expenses to be defrayed by us or our representatives, if occasion arises and the matter is not otherwise arranged for.

“Thinking it unnecessary to disturb your esteemed repose, we have proceeded in advance to make requisite preparations, and shall await your respected person at the Green Dragon Inn, Bywater, at 11 a.m. sharp. Trusting that you will be punctual,

“ We have the honour to remain “ Yours deeply “ Thorin & Co.”

“That leaves you just ten minutes. You will have to run,” said Gandalf.

“But—,” said Bilbo.

“No time for it,” said the wizard.

“But—,” said Bilbo again.

“No time for that either! Off you go!”

To the end of his days Bilbo could never remember how he found himself outside, without a hat, a walking-stick or any money, or anything that he usually took when he went out; leaving his second breakfast half-finished and quite unwashed-up, pushing his keys into Gandalf’s hands, and running as fast as his furry feet could carry him down the lane, past the great Mill, across The Water, and then on for a mile or more.

So he is essentially coerced into the adventure by Gandalf.

He goes out of a sense of duty and, over time, finds himself an integral part of the group and actually enjoying it. The reasons for this enjoyment, as Gandalf knew, were also touched on in the first paragraph:

Still it is probable that Bilbo, her only son, although he looked and behaved exactly like a second edition of his solid and comfortable father, got something a bit queer in his make-up from the Took side, something that only waited for a chance to come out. The chance never arrived, until Bilbo Baggins was grown up, being about fifty years old or so, and living in the beautiful hobbit-hole built by his father, which I have just described for you, until he had in fact apparently settled down immovably.