Actually, the technology isn't that new. It seems however, that the setups are becoming more affordable in the hobby market.
The technology is know as stereolithography and is a form of light-induced polymerization (UV curing).
The liquid monomers used have a substituent that can be activated using UV light. Out of practical considerations, UV is more convenient: polymerization of the resin will not occur under natural light.
In principle, the procedure is comparable to UV curing in dental repair. There, a lamp that irradiates the whole area is used: you want a fast hardening all over the tooth.
In 3D printing, this is obviously different! The light source is a laser with a narrow beam: you only want to irradiate a very small volume element at a time.
You don't want a rapid polymerization either: only the volume element irradiated should react!
The laser beam is focussed at the boundary of the resin and different volume elements are addressed by moving the focus in the $xy$ plane.
Rather than moving the (heavy) laser itself, the beam is "moved" using a mirror optic. This allows for a very fast adressing of reaction points within the plane.
In the example linked, in the question and a comment, however, the setup is different from laser-based STL printers.
Irradiation from the bottom of of the resin tank through an oxygen-permeable membrane indicates indeed that the UV curing proceeds in the presence of oxygen. Anyway, if the light source would be just a lamp, one would nevertheless achieve homogenous curing of the whole layer.
But the manufacturer doesn't use the term lamp or light source. Instead, they write about a projector and that's probably what they do:
They are not illuminating the whole layer, but are projecting masks for every slice of the object to be printed.