In a nutshell, evaporation is a cooling process. The alcohol has a relatively high vapor pressure (compared to sweat - salty water). The warm body causes the alcohol to evaporate even more rapidly than usual. The alcohol is a liquid and the heat of the body provides the energy for it to evaporate. This is an endothermic process and heat always goes from a warmer area to a cooler one. The body heat is carried off to the cooler atmosphere by the rapidly moving alcohol molecules as they evaporate.
