Actually saw a video once where a guy explained how pumps work. They move fluid by creating a difference in pressure, not just pushing air. The impeller or piston physically displaces the water. Calling it just air flow misses the whole mechanical action. Pretty basic fluid dynamics, really.
Funny you mention that @rose_allen60, I had a buddy who swore a shop vac could drain his pool... until he actually looked at the specs on a real pump. That video the_wesley mentioned is spot on though. I messed around with a small submersible pump in a bucket once, and seeing the impeller spin made it click for me. You can feel the water getting pushed, not just pulled by some air trick. It's one of those things where the simple explanation works way better than trying to sound clever.