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There is a difference between evaporation and boiling,for example the boiling point of water is 100 C degree, but water does not have to be heated 100 c degree before it turn into gas. After a downpour of rain, the raindrops eventually dry up even though the temperature is much lower than 100 C. we say that the water has become water vapour in the air. This is the process of evaporation. We can think of a vapour as a gas at a temperature below its boiling point. 
A liquid evaporate more quickly as its temperature approaches its boiling point. That is why raindrops disappear quickly after a storm, where the temperature may be 30 C degree, but they may lie around for days in a cold region where the temperature is close to
0 C degree. 
Imagine a beaker of water. The water will gradually evaporate.

The particles of water are moving around, and some are moving faster than others. Some may be moving fast enough to escape from the surface of the water. They become particles of water vapour in the air. In this way, all of the water particles may eventually escape from the beaker, and the water has evaporated.

If you get wet, perhaps because you are caught in the rain or you have been swimming, you will notice that you can quickly get cold. The water in your body is evaporating, and this cools you down.
So why does evaporation make thing cooler?
Because the particles that are escaping from the water are the fastest-moving ones. They are the particles with the most energy. This means that the particles that remains are those with less energy and so the water is colder.