This. IANAP (...Not A Physicist), but condensing water vapor releases a whole lotta heat. If you could do that "free" - well, setting up a heat engine between your hot condenser and the evaporation-cooled spot where you let the water vaporize again becomes a classic perpetual motion machine.
(Yes, I see that the article claims their system is a mere 10 to 30 times more energy efficient than conventional dehumidifiers. That still sounds far too good to be true.)