Right so looking at the paper here and the proposed mechanism, it looks like basically they've built a plate-less super-capacitor. In short; a normal capacitor

Right so looking at the paper here and the proposed mechanism, it looks like basically they've built a plate-less super-capacitor. In short; a normal capacitor is two conductive plates with a dielectric separator. In a super-capacitor you have an electrolyte and a porous membrane which can absorb it to allow the ions to build up an even bigger charge separation layer. It seems like what happens here is that their porous material does the same thing, but the "charge" is the incidental charge on the ambient humidity in the air. The nano-pores are basically small enough that you make the likelihood of a being able to strip that charge from a collision very high, and thus get a current as you discharge the capacitor. Which answers one of my questions: no this doesn't actually remove humidity from the air, so that's disappointing but expected. But my second question is whether what they've really found is a new class of super-capacitor separator materials - since you'd get a lot more current if you intentionally jammed a lot of charged ions into close-proximity. Or at least a mechanism by which these can be optimized.