That voltage from the crystal creates an electric field, which exerts a force on free charges in the air. (There's always some free charges floating around, like electrons and ions.) These charges accelerate and collide with air molecules, kicking loose more free electrons that also accelerate and collide, and so on, in an exponential chain reaction. We call this an electron avalanche. And guess what happens? Accelerating charges create a changing electric field—which, yup, creates a changing magnetic field, etc., and that disturbance radiates outward as an electromagnetic wave. Huh? It has nothing to do with "free charges in the air." That's... kind of the whole idea behind EM theory.
That said, I've never heard of anyone building a detector out of balls of aluminum foil, so that's pretty cool. I'd classify it as a rectifying detector rather than a coherer, though, because nothing is physically moving. A real coherer had to be physically bumped or tapped between received signals.