Following on what @Oofta wrote, D&D was created as an amalgam of ideas from other media forms, including Tolkien. In Middle-earth the giant eagles were actually mystical beings, even quasi-deities (I can't remember if this was speculation or Tolkien himself, but they may actually have been Maiar--that is, lesser angelic beings/demi-gods, of the same ontological order as the Istari and Balrogs). They were, in a sense, the embodiment of an archetype of the natural, animal world, of which dragons were the twisted form.Now none of this "deeper stuff" was explicated by Gygax, as far as I can tell. Gygax and the designers of D&D drew upon many sources and assimilated them into D&D for game purposes, so the metaphysical or archetypal stuff wasn't really part of that conversion - except perhaps through implication. But I think the Tolkien approach works well: imagine them as quasi-immortal, mystical beings that are "guardians of nature."
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