Why would lower pressure cause water vapor to condense?

Actually, that hasn't anything to do with how air conditioners work. Air conditioners do not use air to produce a low temperature, they use a throttling process of refrigerant to do that. Drop the pressure of liquid refrigerant by sending it through a flow constriction, and it cavitates to a two-phase liquid/gas at a lower temperature and pressure. Let the intake air finish boiling the refrigerant, compress the refrigerant to high pressure&temperature, and then chill the refrigerant back to a liquid against outside air. THAT IS HOW AN AIR CONDITIONER WORKS. Lower pressure itself doesn't cause water vapor to condense. If you change pressure isothermally (that is, at constant temperature), it would do exactly the opposite...cause water to evaporate. What MUST happen is that you change the pressure ADIABATICALLY, that is, without heat transfer. All of the work done by the gas as it expands (done on surrounding air, pushing it out of the way) comes from thermal energy of the gas. This in-turn decreases its temperature, and decreases its temperature SO MUCH that the water will prefer to be liquid, even once at the lower pressure. This is part of the process of cloud formation.

When there is a low pressure area,winds from the surrounding areas converge there and are forced to go up.This rising air expands as it goes up causing cooling of the air(normally expansion of a gas causes cooling).Moreover the temperature normally comes down as we go up.Hence the air is brought down to its dew point temperature causing condensation.