How does air blowing from a fan make you feel cooler if the air is the same temperature?

Air moving across your skin promotes evaporation of the moisture on
your skin, and evaporation always carries heat energy away with it.

That's the whole purpose of sweating ... Your insides say to each other:
"We're burning up in here and we've got to get some relief !  Pump that
moisture out to the skin where it can evaporate and carry away some of
this heat energy that's building up in here!"

That's why you feel even hotter on a very humid day ... because you're
not able to evaporate as much moisture into air that's already very humid,
even if it's moving.