It won't (my personal totally empty prediction). To get humidity in larger scale it means getting air flow. Air flow in sufficient numbers doesn't come out of no where. Usually with this kind of stuff one quickly finds one needs just insane amount of flow to scale meaning big blower fans and then you find you spend all the energy you produce running the fans and other needed ancillary equipment.

“The beauty is that the air is everywhere,” says Yao. “Even though a thin sheet of the device gives out a very tiny amount of electricity or power, in principle, we can stack multiple layers in vertical space to increase the power.”

On the first sentence... Sahara and Gobi would like to disagree on the tech working everywhere.

And when you start stacking the layers you need pressure to push the air through the layers and so on meaning, supply fans. Otherwise eventually on big enough system you find.... the system sucks all the moisture out of the air locally and then no more electricity. They aren't pulling energy out of air, but out of humidity . First is plentiful in atmosphere, second is at times very finite quantity locally. Sure on the wider atmosphere the humidity is plentiful, but again how you get that humidity to the device constantly. There needs to be airflow. With small enough device like those micro watts, well the humidity present ambiently is enough, since it consumes next to nothing. Start to scale up and well the ambient humidity is not enough. Not unless you are at windy sea front at which point.... why not just put up a wind turbine and a sea front sea wave power station.

It might find utility as small local power source with not much power required, but grid scale thing it most likely won't be. aka it isn't hog wash, but when they start talking "yeah, but we put 100k of these disks in a stack and it will be this much power" you must start asking "so how many kilogram of H2O is that thing ingesting per minute, is there that much H20 weight in the air in the first place. If not how big fans and turbine you need to drive new moist air to it".