One of the most clear and evident things about the various races, creatures, and foul monsters of Middle Earth is that there is often a fine line between them. In fact, many of the cruelest broods of the world were corrupted from once better things. This can be seen in the examples of the nine Ring Wraiths, who were once great and noble lords of men; the Huorns, who were once peaceful trees awoken and taught to speak, now turned vengeful and wicked by their mistreatment; and the Orcs, who are thought to be twisted versions of elves with tortured souls from the malice of Sauron’s will.
Other evil creatures from Middle Earth have sprung up from the darkness, appearing exactly as they are now, rather than evolving from a once good being into something more foul. Good examples of this include the Watcher in the Water outside of Moria, a being that seems to have surfaced from some deep, cavernous ocean long ago, drawn by the growing evil in the world; Shelob the spider demon, who has lurked in her lair for thousands of years since her mother Ungoliant first birthed her; and the Balrog of Khazad-Dum, who attacks the fellowship on their journey underground, and manages to take Gandalf the Gray with it when it falls.
But one creature that is hard to categorize as either of these types are the Fell Beasts, the flying servants of the Nazgul. Tolkien never specifically delves into where they come from, other than hinting that "A creature of an older world maybe it was, whose kind, lingering in the forgotten mountains cold beneath the moon, outstayed their day."
Very little is known about them in general, but some fans have followed along the train of thought of the orcs who were once elves, the wraiths who were once men, and even Gollum, who was once an old stoor-hobbit. This theory posits that the Fell Beasts may actually be Giant Eagles, who have been poisoned by Sauron’s evil and infected under his control, turning them into the reptilian creatures over several hundred years.
This theory is difficult to prove or disprove, but the Fell Beasts have no known resemblances to the Giant Eagles, other than their size and the fact that they can fly. They have the fleshy, leathery wings of a bat, as opposed to the feathery wings of a bird though, and have short muzzles full of sharp teeth, rather than beaks. Many fans argue that they are more akin to the dragons of Middle Earth than the Eagles, and that even with the torturing and cruelty that Sauron is capable of, it would be difficult to change their anatomy so far that they don’t even slightly resemble the original creatures they sprung from.
The Fell Beasts are definitely grounded in the real world, very physical, visceral beings, with piercing shrieks and powerful claws, and as can be seen when Eowyn cuts off the head of one, they are possible to kill. But this only lessens their chances of being corrupted eagles. Most wraiths and other corrupted beings have transferred over to the shadow realm, and no longer exist as a physical being, but are rather a sort of spirit, tethered to earthy things by Sauron’s will alone (like the Witch King himself).
Melkor, Sauron’s former master, was the being who first created the dragons, so it is possible that he also created the Fell Beasts in his experiments of trying to make sentient life. Sauron may then have brought one of the eggs across with him after the fall of his master. Alternatively, perhaps Sauron himself wanted to try his hand at creation, after he got a taste for it with the creation of the rings of power. Or maybe they were disturbed and rose out of their hole in wrath and anger when some ancient, unsuspecting culture dug too deep and became too greedy, as all peoples of Middle Earth tend to do at some point.
Either way, with such a world so packed with complex lore and history, many of these things are difficult to provide a concrete answer to. Fans have been coming up with theories to explain things, or add context or depth to Middle Earth for years now, such as Boromir being corrupted by the blood of Sauron from the blade of Narsil, or the famously misconstrued “Fly, you fools,” line spoken by Gandalf. But as Gandalf also said, "There are terrible things in the dark places of this world," and it’s hard to tell where they came from. Wherever that is, there’s one place that they are definitely going, and that’s into the minds of anyone who watches the films. These terrifying creatures are definitely hard to forget.