In a sense, you are right. Your body sweats to cool itself down. The actual act of the water evaporating is what makes you cool. When you are not moving, the air around your body saturates with your sweat, therefore making it harder for the sweat to evaporate, preventing you from being cooled down. When the air is blown at you, it keeps a constant cycle of non-sweat-saturated air circulating around you, allowing the sweat to evaporate, and making you feel cool.