The Eagles are the heralds of the Vala Manwë, and they are in fact either Maiar (beings of the same "race" as the Valar (Ainur), but of lesser stature in terms of actual power) like Gandalf and Saruman, just with the bodies of giant Eagles, or simply eagles that have been raised up in stature and knowledge by the Valar/Maiar.
The Valar are the appointed "Powers" of the world. Men misunderstand them as "gods" but they are actually more akin to archangels, serving the actual God-figure, Eru Ilúvatar.
At the end of the Second Age with the destruction of Númenor, Valinor (the kingdom of the Valar) was physically removed from the circles of the world, and the earth itself was made round. But Manwë commanded the Eagles to stay, so he would always be able to watch over Middle-earth.
That being said, they do not follow any "command" from any corporeal being in Middle-earth (ie Galadriel or Gandalf), and only emerge from their Eyries in times of great need or by command of Manwë himself.