Why does a fan (or any moving air) cool you down?

That's one.

The other is forced convection:

http://en.wikipedia.org/wiki/Convection#Free_and_f...

convection is a modified form of conduction. The immediate process is conduction, but the heated air is moved away and replaced by cooler air. Forced convection can result in 10 times higher heat transfered than with natural convection.

when the sweat evaporates, it takes the heat needed for (latent heat for vaporization) from our body. therefore the body loses heat and gets cooled down. when the air is still, its relative humidity rises as soon the sweat vapours. so the water potential gradient decreases, and the rate of cooling drops. but when the air is moving, (eg: a fan) the higher humidity air is continuously replaced by less ones. so the water potential gradient between the body and the atmosphere rises and the rate of cooling rises. that's why we feel cool when in front of a fan.

The moving air removes heat both by direct contact (conduction) and by evaporation of perspiration.

molecules from moving air absorb more heat from ur skin so you feel cold

It increases the wind chill effect