Here' my attempt at boiling it down into the bare essentials:
Sound travels in air at some well-defined velocity (which is constant throughout the listening space) that depends on atmospheric pressure and temperature. When you add wind to the situtation, sound still travels at the same velocity relative to the air, but the wind is moving the air so the actual velocity that the sound travels is that of speed of sound plus the local speed of the wind (which probably is not constant throughout the listening space). If sound travels by two paths, one of which is assisted by the wind and the other of which is not as assisted by the wind, and they recombine elsewhere with one of them arriving a half wavelength sooner, then they will recombine destructively and the combined volume will be reduced. When the wind changes and doesn't give the wind assisted path that extra 1/2 wavelength of extra distance traveled, then sound arriving through those two paths recombines constructively and the combined volume will be increased.
Boiled down even further: Wind simply blows the air around while it carries sound on its way to your ears. The resulting sound that you hear gets to your ears by different paths each time the wind changes.
Boiled down even further: Wind blows the sound around and makes it sound all swirly.