The dwarves were captured and imprisoned by the elves in Mirkwood. Bilbo had to keep entering to form a plan and carry it out.
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He was to burgle the Arkenstone from Smaug. Until that happens he is still a burglar.
First of all, he had to get food to eat. Then he used his skills to get the keys and release the dwarves from their cells and help them escape.
The dwarves were captured and imprisoned by the elves in Mirkwood. Bilbo had to keep entering to form a plan and carry it out.
Bilbo Baggins is loyal because when the dwarves were captured by the goblins he went back into the horrible, horrible tunnels and look for his friends. He had just made up his mind that it was his duty, and that he must turn back.
Bilbo wants a looking glass (a mirror) in order to see himself in the elvish armor he attains from Thorin in the old dwarvish palace under the mountain. He thinks he must look rather ridiculous as far as hobbits should look, but still has a desire to see himself.
"No!" said Thorin. "There is more in you of good than you know, child of the kindly West. Some courage and some wisdom, blended into measure. If more of us valued food and cheer and song above hoarded gold, it would be a merrier world. But sad or merry, I must leave it now. Farewell!"
This is a slight oversight in the movie. The Weasleys continue to live at the Burrow, it must have been repaired somehow as they continue to live there later. In the book it wasnt destroyed, at least not to that extent.
Bilbo can see that he and the dwarves are about to be overwhelmed by the spiders and decides he has to disappear in front of them. After they get away the dwarves are bewildered by his turning invisible and must have the story of his escape from Gollum again. Balin in particular feels he is owed an explanation because Bilbo made him look bad earlier when he snuck past Balin on guard.