When someone sweats, that phase change that occurs is evaporation - the sweat evaporates from the person's skin and lowers the temperature of the skin.
Evaporation is a endothermic process, meaning that energy is absorbed. As the water molecules on the skin absorb heat energy from the skin they vibrate faster. Eventually the vibrational motion overcomes the forces of attraction between the molecules and they change from the liquid phase to the gas phase. The molecules that don't evaporate are now cooler because the average kinetic energy (energy of motion) is lowered by the faster-moving molecules leaving.
You've probably noticed the same effect when you get out of a swimming pool on a windy day. Wind speeds up evaporation, so you feel colder than you would without wind.
There's a limitation to sweat being able to cool people. The amount of water than will evaporate is related to the relative humidity. In very humid conditions where the atmosphere is nearly saturated with water vapor evaporation is greatly reduced. This is why a dry 90 degree day is more comfortable than a humid 90 degree day.