A more 'nuts and bolts' explanation is as follows. Molecules in the water have a range of kinetic energies. The faster molecules on the surface have enough energy to take them up into the surrounding air and to become part of the surrounding water vapour.
It is the Mean KE of molecules in the water that determines the Temperature and, having lost some of the faster molecules, the mean KE is reduced - hence the temperature drops.
Once the air near the surface is 'saturated', there will be as many molecules returning (adding their KE to the rest) as leaving so net evaporation will be zero and the temperature will cease dropping. This loss and return of molecules is often described in terms of 'vapour pressure' (changes of Momentum on the surface).
When there is a current of unsaturated air over the surface, the escaping molecules get swept away from the surface and so more evaporation will take place, reducing the temperature further (e.g. on the washing line on a windy day or standing in a wind after a swim).
This explanation does away with the notion that air is, somehow, like a sponge - absorbing the water. The molecules of water vapour are in no way different from the O2 and N2 molecules also in the air; all three gases exert their own pressure and add it to the final pressure in the air. The same vapour pressure would exist over the surface of water if it were in a container which had been initially evacuated.