What process allows perspiration to cool the body?

Evaporation.

Sweat produced by our bodies during exertion coats the exterior skin surface. It is the evaporation of the water in sweat by the local conditions of temperature, humidity and moving air currents that cools the body.

Water is 4.178 J/g-'K. Its mass can be taken as 1.0g/##cm^3## for this example. The heat of vaporization for water is 2257 J/g. So, every gram (about 1 mL or ##cm^3##) of water evaporated from your body removes 2257 Joules of heat from your body (neglecting possible losses also from raising the sweat temperature before it evaporates - the specific heat value).