How does this apparition happen?

It wasn't explained in the film but from the context we have a couple options:

1. Snoke is lying. He trying to make Rey feel like she has no control, that her seemingly emotional connection with Kylo was all a farce, and she is destined to turn to the dark side. This would fit well the deceitful nature of the dark side, and plus Snoke has no real incentive to tell the truth.
2. Snoke was telling the truth, but once Kylo and Rey's minds were connected they stayed connected. Snoke opened the door, they are just walking through it again.

Kylo and Rey are pretty powerful force users. I'd say once they were aware of the ability there is no reason why they couldn't use it on their own.

Bonds are entirely between the two parties involved in the bond. It's unclear how Snoke caused the bond between Kylo and Rey, but the rest of the bond seems to fit with the only other canon appearance.

In the Clone Wars episode "Sacrifice (S6E13), Yoda is shown to have a bond with Dooku, since they had been close as a master/padawan. This bond also allows emotional sensing between the parties, even if one does not wish it (as we see in The Last Jedi, on Ahch-To).

Again, it's unknown how Snoke bonded Kylo and Ren without them knowing (until it was done, obviously), but it's definitely the same kind of thing as the standard force bond.

To quote Skooba's answer, "once Kylo and Rey's minds were connected they stayed connected".

My take on it was that the earlier bonds/connections, where Rey and Kylo could see and touch one another, were facilitated by Snoke while he was still alive.

The one connection that happened after Snoke had been killed, which took place just before Rey closed the door of the Falcon, I presumed to be a simpler connection where they just mentally communicated with each other (ie. Force telepathy), very similar to how Luke and Vader communicated at the very end of Empire.

Yes, the bond still remains. Only logical explanation that anyone can think of for this is "FORCE". But why does it still remain after Snoke died?

I think it was a plot device for future interactions , Because Snoke died Kylo has no one to interact other than the generals, this makes his scenes boring in next movies. With Rey still being able to contact him can be used for some drama.

"I've got a bad feeling about this"

I had a few ideas on this.

1. I thought it was too obvious, but they must have gone down the twins route. Luke and Leia in the originals have this bond to more of an extent than anyone else seen in the movies. Snoke knew this and attempted to use it to his advantage. Which is why the bond stays around after he is dead.

2. They are the last Jedi's, and therefore there is a strong bond between them allowing them to stay connected.

3. Snoke isn't dead

I think Disney has been doing a pretty good job so far on their story, I doubt it would be something obvious and certainly hope it isn't because they are related. The whole idea of anyone being a Jedi is great.