It said Bilbo disappeared after his and Frodo's Birthday Celebration together. Where did Bilbo leave to and why did he leave?
The why is directly addressed at the start of The Fellowship of the Ring. In short, despite merely bearing the Ring of Power (rather than wearing it regularly) Bilbo has begun to feel its effects. This seems to be proportionate to the growing power of its true owner, Sauron.
He is afflicted with wanderlust and feels that he wants to escape from his life, not realising that what he actually wants to escape is, in fact, the ring that he always carries with him.
He's also justified the decision to himself that the presence of his relatives has distracted him from the completion of his magnum opus, the Red Book.
'Yes, I am. I feel I need a holiday, a very long holiday, as I have told you before. Probably a permanent holiday: I don't expect I shall return. In fact, I don't mean to, and I have made all arrangements.
'I am old, Gandalf. I don't look it, but I am beginning to feel it in my heart of hearts. Well-preserved indeed!' he snorted. 'Why, I feel all thin, sort of stretched, if you know what I mean: like butter that has been scraped over too much bread. That can't be right. I need a change, or something.'
The where is also addressed. Bilbo's original plan was simply to strike out for the mountains and to travel. Gandalf suggests that he goes to visit the elves.
Gandalf looked curiously and closely at him. 'No, it does not seem right,' he said thoughtfully. 'No, after all I believe your plan is probably the best.' 'Well, I've made up my mind, anyway. I want to see mountains again, Gandalf, mountains, and then find somewhere where I can rest. In peace and quiet, without a lot of relatives prying around, and a string of confounded visitors hanging on the bell. I might find somewhere where I can finish my book. I have thought of a nice ending for it: and he lived happily ever after to the end of his days. ' Gandalf laughed. I hope he will. But nobody will read the book, however it ends.'