Triple cooked chips: Removing moisture by cooling
There are a lot of things going on to change the starches in the potatoes besides drying too; see rumtscho's answer for more on that. However, the drying matters as well, and these are indeed good methods of drying.

Refrigerators and freezers are notoriously good at drying things out. You might've noticed food drying out in your fridge if it wasn't well-covered, and freezer burn is a huge deal.

The air inside fridges and freezers is kept pretty dry, and it's circulating, pretty much ideal conditions for drying. Freezers really need to be pretty dry, since they're trying to prevent frost formation. But fridges are dry too too: the air is only a little above freezing, and the cooling coils are below freezing, so moisture is drawn out. Either way, it's still air-drying, just at a cool temperature.

Hot air is also good at drying things out, but I don't think you actually want the drying stage to further cook your food. Leaving it at warm room temperature with a fan would dry it pretty well too, but then you'd have a food safety issue.

The main alternative would presumably be a dehydrator, running at a bit over 140F, thus keeping things safe while avoiding further cooking. But that wouldn't accomplish the same changes in the starch, and that's not something most people have, so the fridge/freezer seems to make a lot of sense as a common, effective method.