Because evaporation is a cooling process. As Walter Vink of Vink Associates, a New Jersey consulting firm to the pharmaceutical and confectionary industries, points out, mints contain sugar-alcohols like the menthol in peppermint, for example that evaporate when they hit the warm, moist surface of your mouth. It's similar to how sweating helps cool your body in the heat.
Sugar free mints work a little differently. They have what's called a negative heat of solution, meaning that when the candy dissolves in your mouth, it actually absorbs heat. With both types of mints, the temperature drop, or cooling, is very small. The sensation is quite noticeable because your mouth is loaded with nerve endings.