When Bilbo, the dwarves, and Gandalf were trapped up in fir trees with wargs (evil wolves) and goblins beneath them, the eagles (including the Lord of the Eagles) swept down from the mountains and carried the Company to safety in their eyries.
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The eagles hate the goblins and wargs, so are more than willing to do anything that would annoy them! In this case, they rescued the party to prevent the dwarves from having their fun, more than any desire to save the group. It is only afterwards that Gandalf and the eagles discuss a way to help the group get where they are going.
They needed to be rescued by the Eagles because they were up in the trees and if they had jumped down they would've been attacked by Wargs and Goblins and the story would've ended there in the middle of the book. The only way to get out of their situation was up. And as the dwarves, Bilbo, and Gandalf couldn't fly, they got the help of The Eagles. As The Eagles are Gandalf's friends and also the Eagles keep the goblins from attacking innocent men villages.
He devises a plan to get them out of the dungeons. A stroke of luck allows him to get the keys from the jailor. He then puts the dwarves in barrels that are dropped into the river to go back to Laketown.
He stole the keys and let them out of their cells. Then he packed them into barrels and they floated to freedom.
The eagles snatched the party out of the burning trees. They disliked goblins and wargs and were always willing to annoy them.
He was the first one to step out into the open, into the elves' light, and so the magic hit him hardest, and he fell dazed to the ground. The wood elves carried him off with them when they ran away, but the other dwarves and Bilbo were left behind.
Bilbo stays invisible for weeks listening to conversations of the Wood-Elves. The king's first man<-? and a guard test some wine in the basement of the Wood-Elf compound. They fall asleep and Bilbo steals the guard's keys. Bilbo unlocks the cells of the dwarves and he puts them in empty tubs that the Wood-Elves were about to throw out through their river disposal system, (I can't quite remember what it was called), and the elves who throw out the tubs threw out the dwarves without even looking in them. Bilbo had one problem though, he wasn't in a tub, so he had to grab on to a barrel and float into the lake where the dwarves and Bilbo are accepted and fed and housed.
well
The reason why Thorin didn't want the elvenking to know about their quest is because the wood-land elves are selfish and love gold, the dwarves are trying to get their gold back, but if the elves find out, they will want to help them, but get some of their treasure!
Yes, in chapter 9 the dwarves (but not Bilbo) were imprisoned by the Wood-elves in Mirkwood forest.