Basically this is just an aluminum plate with fins, glued into the bottom of a sealed cardboard box. A water tray is also glued in, next to the plate. Paper towels lay across the plate, and drape into the water tray, wicking water up onto the top of the plate. The fans in the box lid blow air directly on the paper towels, evaporating the water and sucking heat out of the plate, and the humid air is exhausted out the side of the box. A third fan is placed under the box (propped against the bricks in the pics), and blows air from the room across the fins which are protruding out the bottom of the box, cooling the air.
To see first hand how this works, grab a can of your favorite beverage that is at room temperature or above. Wrap the can in a paper towel, and get the whole thing wet. Place the can and wet paper towel in front of a fan, and check the temperature of your beverage when the paper towel is almost dry.