The title seems pretty straightforward: the story is called "A Very Old Man With Enormous Wings," and we do, indeed, get the story of a very old man with enormous wings. But this title tells us something important. In the old-man-versus-angel debate, the story comes down on the side of the old man, not the angel.
But even here this is kind of a joke. The story is telling us that the supernatural explanation isn't right. This is no angel, just a very old man. With enormous wings.
See? There we are, right back at the supernatural. The straightforward doesn't seem so straightforward anymore.
And the title gets another layer of complexity with its subtitle, "A Tale for Children." "Come on," you're thinking. "There's no way this is a story for children."
And you're right. The title's irony cues us to think that there's something more going on in the story—at the same time that, by telling us it's "for children," it winkingly suggests that it's just a simple tale.