The answer everyone missed: boundary layer. Air has friction. When you sit there, a boundary layer of much slower moving (relative to the motion of unobstructed air as connection occurs) air creates an insulating effect around you. The fan removes the boundary layer by blowing it away.

This effect is why food that is boiled cooks so much faster than food at 99C. You’d otherwise expect it to cook only a little over 1% faster but the actual effect is far greater. The boiling churns the water and disrupts the boundary layer. Similar principle in a forced convection oven.

A fan works by speeding up the rate of evaporation from your skin, which cools you off. At these wet bulb temperatures the ratio of heat/humidity is so high that blowing hot air over your skin just heats you up faster. Kind of like turning on the convection feature in an oven.

Yes, exactly. If you pour water on you and sit in front of a fan, why the windchill effect wouldn't work anymore?!

I think this stuff only happens in absence of wind...

Because the temperature we're talking about is the wet bulb variety. That means it accounts for evaporation. What is normally a windchill is now a "windwarm" effect.

(Yes, I know that windchill effect isn't solely due to evaporation-cooling. It's also stripping the layer of warm air away from you.)

The environment is hotter than you. Blowing hotter air (laden with moisture, so it won't evaporate your sweat) won't do anything to cool you down.