"They're a variation on a swamp cooler that takes in air, evaporates a 'lot' of water into it, then uses a heat transfer system to use that air to cool the hot air in the building."
The heat transfer mechanism in a swamp cooler is the evaporation cycle.
Hot dry air hits the exchange medium, which causes flash evaporation of the water and drops the temperature of the air that comes out the other end. The output air is of higher humidity, but in areas where swamp coolers are usable, that's not such a bad thing. The efficiency and effectiveness of swamp coolers drops dramatically as the ambient humidity increases.
"It is hot as heck, of course, so the water evaporates, but the evaporation actually helps cool the unit more, which helps it be a touch more efficient."
That's how virtually all window air conditioners now work -- the water that is condensed on the interior coils is transferred to the exhaust side and then the exchange fan blows some of it onto the hot coils while the excess drips out, giving a small patch of lush green grass below the window.