why do the dwares finally agree to bilbo's escape plan

When Bilbo first tells the dwarves his plan they are completely against it and complain:

"Bilbo saw that the time had come to explain his idea, as far as he could; but he did not feel at all sure bow the dwarves would take it. His fears were quite justified, for they did not like it a bit, and started grumbling loudly in spite of their danger.

"We shall be bruised and battered to pieces, and drowned too, for certain!" they muttered. "We thought you had got some sensible notion, when you managed to get hold of the keys. This is a mad idea!"

Bilbo's response was one of pure common sense:

"Very well!" said Bilbo very downcast, and also rather annoyed. "Come along back to your nice cells, and I will lock you all in again, and you can sit there comfortably and think of a better plan-but I don't suppose I shall ever get hold of the keys again, even if I feel inclined to try."

Source(s): The Hobbit