It's not so much that they saved the day; they just joined the battle at a fortunate time. In the book, the five armies were (in order): Elves, Men, Dwarves, Orcs/Goblins/Wargs/etc., and Eagles. The Eagles generally don't get involved in other's affairs, but they hate goblins and orcs and usually thwart their plans when they can. They didn't join the battle to help the "good guys" necessarily, they joined because they had followed the movements of the Orcs and knew they were up to something.
In the book, Tolkien says that the battle started to look grim, but the arrival of the Eagles helped push the direction of the battle in the favor again.
Edit: For reference, Peter Jackson really overemphasizes the effect of the Eagles in his movies. They helped, yes, but he makes them look like the savior of every scene they participate in.
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