Air is blown through the material by a regular fan, which results in the evaporation of the moisture and the creation of a cooling breeze. This breeze can reduce the air temperature n a room effectively by thirty degrees or more. Since there is no refrigeration process which exchanges the hot air for the cold, then there is no exhaust air requiring venting.
Swamp coolers actually blast room temperature air through a wet membrane, resulting in cold air. It doesn’t require power to operate as it has no other mechanical works, making it much cheaper than a regular air conditioning unit.
This is therefore how they work, and what contributes to them actually being truly portable in every sense of the word.