Water evaporating absorbs energy - the transition from liquid to gas takes more energy than changing the temperature of the water, so drying the clothes is a net reduction of energy in the surrounding air.
It relies on the air not being saturated (and for real effectiveness, requires relatively low humidity). It's quite popular in dry climates in the form of a "swamp cooler".
https://en.wikipedia.org/wiki/Evaporative_cooler