Why will placing a damp towel over a fan increase its ability to cool a room?
Water has a relatively high volatility, which is the ease at which a liquid evaporates. Therefore, the water needs energy to evaporate. Where does it get the energy? It gets it from the surroundings.
The energy needed for the water absorbed (this goes along with volatility which is the ease at which a liquid evaporates) on the towel is high, so it will need to absorb energy from its surroundings in order to evaporate. This will decrease the temperature in the room,
However, what you need to consider is how the evaporated water will effect the humidity of the room. If someone is sitting in this room and sweats, because of the humidity in the air caused by the damp towel, the rate of net sweat evaporation will be the same but the rate of condensation of sweat back on the skin will be higher-- this means the person will sweat more even though the temperature in the room is lower. So this process is very similar to the process of how sweat cools you down.