How does 3dCarbon CLIP printer work?
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.


UPDATE
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.