I think the answer is that a lot of birds are real, but most are government spy robots. It's been a long time since I visited the Birds Aren't Real website, though, so maybe I'm wrong or maybe their explanations have changed.
Right. This is because, empirically, I can see when I shoot a bird that it is, in fact, a real bird. However, that's only because the gov't birds are smart enough to avoid my hunting grounds and are likely impervious to my shells, anyway. That's how I make sense of it, at least.
I'm very interested to hear more about mocat's theory, though. Color me skeptical.