Is it me only seeing a small hot wire as a razor? So the guy is trying to avoid touching it to his skin? Seems like not laser at all. The other thing is that the wire seems to be hot after touching the hair, so you can see it glowing.
It is a glass optic fiber suspended across which forms the blade "edge". When light enters a fiber like this it bounces off the internal walls and travels along the glass fiber without exiting... Otherwise you would have all your light escaping out the sides very quickly.
It just so happens that when you have AIR and GLASS interface, the incident angle for a beam to hit the inner side of the glass/air interface has to be greater than some angle for the light to pass through. Otherwise almost all of it gets reflected back. But as soon as you change the materials that make up the interface (say make it GLASS/GLASS) the light can pass right through.
Imagine you have a piece of GLASS with light inside. It is surrounded by AIR. Light hits the inner wall of the glass (to try and escape) and bounces off. Because the GLASS/AIR interface requires a larger angle to escape than what the light hit at.
Now put another piece of GLASS up against the first GLASS with light in it. As far as the light is concerned, it has no idea it left one piece of glass and went into another. If there is no air space between them, the light thinks it is in the same piece of GLASS and will leave the first piece and enter the second piece at pretty much the same angle.
This is all basic stuff about reflection, refraction and what's called the critical angle.
Anyways, the "Skarp" works because when HAIR touches GLASS, the light is no longer interfacing a GLASS/AIR interface (which causes the critical angle to be such that light bounces back into the glass fiber). When you have GLASS/HAIR interface, the critical angle changes so that the light can now escape and travel through the hair, causing it to burn to a crisp.
That's the idea behind the technology. It is nice in theory, but will take WAY WAY more refinement for it to work. For one, they need to create a SOLID FIBER channel rigidly kept in place and not a thin piece suspended in thin air which will easily break. Also, it will need to bend a little to allow some flex over curved skin surfaces. It will also perhaps need some forced-air to come out the tip to blow away the hair as it cuts, so it doesn't "fuse" back and just dangle half-cut like it shows in the video.
Anyways, LOTS of refinement is needed... They are going to burn through that $4 million fast.
(EDIT: Spelling mistakes)