I haven't looked at the site very deeply, however, upon a first look, it appears that they are extrapolating a known phenomenon (Vitamin D light therapy and some skin conditions) that has a sound scientific basis, and extrapolating it to a whole host of unrelated benefits and applications.
This is the standard mode of operation of most pseudo-scientific claims out there. They will take a phenomenon that the general public doesn't understand that well, and make claims that sound somewhat related to an actual benefit, and over hype it... So going from accepted treatments, to promoting wound healing and pain relief is a bit of a stretch beyond what the established benefits of light therapy are. Of course, the UV light a inhibit some bacterial growth in a wound, so if you are prone to getting red and inflamed wounds, it may indeed help, but not because that's the main purpose of these devices, but just aids in a process that someone may otherwise neglect (i.e. keeping wounds clean).
The claim that this helps with burns is especially dubious. UV radiation is the main component that helps develop a sunburn. The claim that it helps with pain management is also very dubious. Light is absorbed by the skin, and penetrates at most only a fraction of a millimeter. Any actual effect on deeper structures in your body is very dubious at best.
I'll note they give an impressive sounding list of documents to support their claims, until you look at it more closely. The studies that support some of their dubious claims are "unpublished" or "pilot studies". Furthermore, I would be interested in exactly what those studies say. Just because a study has a title that supports a product, doesn't mean that the content of the study bears out with what the product promoter says.