How does 3dCarbon CLIP printer work? Recently there has been announced a new 3d printing technology which uses UV light, from lasers or a projector, to harden slices of epoxy resin so that the printed object rises out of a pool of epoxy.

There is some information about an oxygen boundary and UV light, can someone explain the chemistry of the process?
https://www.youtube.com/watch?v=UpH1zhUQY0c
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.