> This can be done by storing energy in the material itself (that's how desiccants work)

This is exactly where the energy goes. From the paper (in it's Materials and Methods section) -- All measurements were performed at 20° ± 0.2°C maintained by an air circulation system unless otherwise noted. The temperature of the films was controlled using a heating/cooling unit (THMS350V, Linkam Scientific Instruments, Salfords, UK) when necessary.

So the hypothesis is that the heat in the water vapor goes into the nano-pore material, which in their experiment they were actively maintaining at 20 degrees C. So yes, they are actively removing the heat created by the phase change.

One difference with desiccants is that once they are saturated you have to restore them through heating them up, but this stuff doesn't have that property. And while it may sound like nonsense it was reproduced in another lab[1].

Apparently capillary condensation is a thing, its the popping out of the liquid water that was unexpected.

[1] With a material that could potentially defy the laws of physics on their hands, Lee and Patel sent their design off to a collaborator to see if their results were replicable.