Evaporation is said to cause cooling effect because it absorbs energy from surroundings to change its phase from that of a liquid to gas. I am in doubt as to why would the surroundings be ready to give up energy more than the water molecules. Wouldn't 'lower temperature of the water molecules in comparison to the surroundings' or other such conditions which might involve specific heat capacity or conductivity etc. be mandatory to ensure that the heat is transferred from the surroundings to the water molecules and not the other way round ?
Evaporation requires the liquid molecules to overcome the intermolecular attractive forces and escape to the surroundings. Only the most energetic molecules, near the surface of the liquid, have enough kinetic energy to overcome the attractive forces. As these highly energetic molecules escape from the liquid (i.e., evaporate), the average kinetic energy of the molecules in the liquid decreases and temperature is nothing but a measure of the average kinetic energy of the system. Hence, temperature decreases with evaporation.
Now, for more number of liquid molecules to have energy enough to overcome the intermolecular forces, the average energy of the liquid must be high. The liquid can gain more energy by absorbing energy from its surroundings. But according to the second law of thermodynamics, this is possible only if the temperature of the liquid is lower than that of its surroundings. Such a flow of energy occurs until the liquid and its surroundings are at the same temperatures (or until all the liquid evaporates).
So obviously, evaporative cooling is possible only if the liquid is cooler than its immediate surroundings. A very good example of that would be sweating, in which the sweat absorbs energy from our body and evaporates.
When you perspire the sweat will be at the same temperature as that of the body. However as the more energetic molecules escape the liquid, the average energy (and hence temperature of the liquid) decreases and this is turn causes the sweat to absorb more heat from the body.