Thorin Oakenshield loses his oaken shield, which he had obtained after defeating Azog in battle, after he loses a battle with Azog. Ready... set... SYMBOLISM!
Azog roars with fury as the eagles carry away the rest of the dwarves, as well as Bilbo and Gandalf. There's little he can do as the eagles soar away, flying throughout the night. The eagles drop Thorin's Company off on top of a huge rock. far from danger, and fly away.
The Book: Gwaihir, Lord of the Eagles, sees the fire from his perch far away. He hears the wargs, and correctly guesses that the goblins are up to some mischief. Gwaihir and his eagles are described as noble and proud, and they hate goblins - they make it their mission to stop whatever wickedness the goblins may be up to, and to drive them back to their caves. The eagles attacked the goblin pack, and snatch up Thorin's Company to take them to safety.
See, it's not just Peter Jackson. This is apparently the only way Tolkien knows how to resolve a climax.
The eagles land at their eyrie, called the Great Shelf. One eagle refers to the dwarves as prisoners, which leaves Bilbo trembling with fear, until he sees Gandalf chatting amicably with Gwaihir. He learns that the eagle referred to them as "prisoners rescued from the goblins," not captives of the eagles. Gandalf, in fact, had once done a favor for the Lord of the Eagles. They were honored guests, and the eagles even brought hares and sheep for the dwarves to cook. The next morning, the eagles refuse to fly the dwarves too close to where men live, afraid of being shot down. Gandalf asked to be taken as close to the Lonely Mountain as possible, so the eagles dropped their passengers off on a great rock several miles away.
The eagles just expected them to climb down, I guess?
What differences does it make? Peter Jackson took several liberties with the eagles. Notice that they are never summoned by Gandalf in the book - there is no moth, and they investigate the fire and goblin's attack on their own. They did not even know Gandalf was present when they attacked the goblins. Further, in the book, the dwarves are taken back to the eagle's eyrie and treated as guests, allowed to eat and rest. They engage Gandalf in conversation, and even explain why they cannot take the Company too far.
My Opinion: Does Peter Jackson just fucking hate Tolkien's intelligent animals or something? First the wargs, then this. The eagles have been transformed from a noble, intelligent race into some birds that Gandalf has trained to come at his call.
It matters. I can't tell you the number of people who have asked, either in this movie or the end of Return of the King, why the eagles didn't just fly them all the way to their destination in the first place. The books give a reason: the eagles chose not to, for whatever reason (it's honestly not too explicit in RotK). The movies just leave the question hanging. Why can't Gandalf's pet birds fly a little longer? They're not shown to have any intelligence at all, or any ability to really make their own decisions. It's as if the movie literally stripped away every example of these birds having some kind of cognition, and I can't for the life of me understand why.
The moth thing was stupid, too. I'm going to just clear this up right now: there is no moth in the books. Any of the books. Gandalf never uses a moth to call for the eagles. In fact, Gandalf never summons eagles. They came to the Battle of the Five Armies, and the Battle of the Black Gate, of their own accord. They came to help Thorin's Company escape the fire and the goblins because... well, they fucking hate goblins. And remember when they came to rescue Gandalf from Saruman's tower? You know, the first time the movies had him use that goddamn moth?
Radagast sent them.
Peter Jackson worked really hard to make Radagast suck so bad.