I watched Lord of the Rings, and I saw a change in Bilbo when he had to give up the ring. He became mad and didn't want to part with it. When Bilbo was giving gifts to Frodo he gave him a shirt. When Frodo unbuttoned his shirt the ring was there on a chain wrapped around his neck. Bilbo told Frodo he wanted to hold it again one last time, and while Frodo was rebuttoning his shirt Bilbo kind of turned into a monster and tried to get the ring back from Frodo. That part scared me, I wasn't expecting that from Bilbo. Why did he want the ring back so badly?
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.
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.