Let's see if I can put this into simple terms
Think of it like this:
Boiling point is a bad word it doesn't mean the point that a liquid turns to a gas but rather the maximum temperature a liquid can get to (at atmospheric pressure) before it turns to a gas.
This is why we fry chips in oil, you can't get water hot enough, get to 100 degrees and it boils and the temperature doesn't go up any higher.
Now that doesn't mean that some of the liquid doesn't turn to gas at a lower temperature - think of all the molecules bumping around in the liquid - some are going slowly, some fast, the very fast ones can get to be fast enough to break free and "evaporate" off as a gas.
Because the faster ones escape like this the overall temperature of the remaining liquid falls. (faster is hotter). That's why evaporating sweat cools you down
There's an animation of the process here
http://www.educationusingpowerpoint.org.uk/ind ex.html?ks3science.html~mainFrame