From the first paragraph of the article:

Even in desert conditions, there exists some level of humidity that, with the right material, can be soaked up and squeezed out to produce clean drinking water.

In the dry dessert the amount if water in the air is so minuscule that it just doesn't doesn't make any sense to do this.

Regarding the "clean" claim, you would get all the crap from the air in your water, also to preserve the water you would need to keep it chilled after boiling it, else you get sick, badly sick.