Why does Bilbo agree to join the dwarves on their adventure?

Bilbo agrees because his character and courage have come into question. The dwarves offended him and made him angry. He agrees in order to prove them wrong.

"Pardon me," he said, "if I have overheard words that you were saying. I don't pretend to understand what you are talking about, or your reference to burglars, but I think I am right in believing" (this is what he called being on his dignity) "that you think I am no good. I will show you. I have no signs on my door-it was painted a week ago-, and I am quite sure you have come to the wrong house. As soon as I saw your funny faces on the door-step, I had my doubts. But treat it as the right one. Tell me what you want done, and I will try it, if I have to walk from here to the East of East and fight the wild Were-worms in the Last Desert. I had a great-great-great-granduncle once, Bullroarer Took, and -"