You guys are making this too hard. It's just pV=nRT (ideal gas
law).

1. Compressor takes in outside air, which typically has water vapor
in it, and packs it into the tank, thus raising the pressure.

2. As you let air out of the tank, the pressure drops. This cools the
air (as p goes down, do does T).

3. Cold air holds less moisture so liquid water condenses from the
water vapor and collects in the tank.

Kevin