The Lord of the Eagles is a classic example of a deus ex machina. When a writer gets himself into a corner, having created a seemingly unsolvable problem, the deus ex machina is the thing that magically comes in to save the day – often out of the blue. In Tolkien's case, the problem he's created is that the dwarves, Gandalf, and Bilbo are all stranded at the top of a bunch of burning pine trees while goblins and Wargs dance at the foot of the trees. How in the world are they supposed to get down? The answer is, they don't go down; they go up. The Lord of the Eagles hears "uproar" in the forest and is "filled with curiosity to know what was afoot" (6.69).
The Eagles are "proud and strong and noble-hearted" (6.68) and not overly-fond of goblins, so when they see the dwarves & co. in distress, the Eagles swoop in and save the day. They do the same thing at the end of the novel as well, with the Battle of Five Armies. When things look to be going poorly for the humans, dwarves, and elves, the Eagles fly in and save the day. So the Eagles definitely keep the plot moving in the direction Tolkien needs it to go. But they don't have a lot of character depth or complexity otherwise.