Actually, they are storks, not eagles. The eagles were wiped out during the War of Wrath when they charged the winged dragons, mistaking them for sparrows--so much for keen eyesight.
Now, because of the confusion during the Dark Years of the Second Age, no record was kept of the extinction of eagles. Therefore, when some mysterious Maia of birth (under which Vala she is remains a mystery) sent her beloved giant storks to deliver babies to the beleaguered cities of Men in the East, people mistook them for eagles. The storks didn't bother to correct them, saying to themselves, "If they be as stupid as to mistake us for anything else but storks, it's their problem."
In due course, they helped the poor people of Middle-earth whenever their busy schedules would let them.
Now, about their flight towards Mordor during the battle before the Black Gate, they were just pushed off-course by strong jet streams when some oaf of a stork weatherman drew the weather charts wrong. Some saw what was happening below them, and decided, "What the Mandos, we're just delivering two babies anyway. Let's just help the poor things."
Of course, if you believe this, I have a piece of Doriath to sell you.
