You are right -- fans actually add heat to a room. One way to think about it is like this: If you have a perfectly insulated room and you put an electric fan in it, then the room will get warmer. All the electricity that is driving the fan turns directly into heat.
So a fan does not cool the room at all. What a fan does is create a wind chill effect.
When weatherpeople talk about wind chill on a cold winter day, what they are referring to is how the wind increases convective heat loss (see How Thermoses Work for details on convection). By blowing air around, the fan makes it easier for the air to evaporate sweat from your skin, which is how you eliminate body heat. The more evaporation, the cooler you feel.
Here are some interesting links:
- How Sweat Works
- How does the wind chill factor work?
- How does a water-cooled air conditioner work?
- How Air Conditioners Work
- How Car Cooling Systems Work