There is nothing particularly revolutionary about this laser. The only unusual feature is the center wavelength (medical applications tend to use infrared). It is more of an engineering feat because it is half to 1/4 the size of similar systems. They do not give detailed specs on their website:
http://www.raydiance-inc.com/productspec.htm
800fs is not a particularly short pulse. Physicists have been using lasers in the attosecond regime for years now. The pulses are so short that they are only a single optical cycle long. However, this company seems to be more concerned with ablation, so pulse energy is important, too. Commercially available Ti:sapphire regenerative amplifiers for modelocked lasers can give you 150fs pulses at the same pulse energy as the one in the Wired article (i.e. peak power levels about 7.5 times higher). However they are huge (take up a whole optical table) and cost about half a million dollars each for a complete system.
http://www.coherent.com/Lasers...ction=show.page&ID=940
Continuum lasers also makes a version of this.
As far as the actual physics of what happens when these pulses are focused onto matter, it is just a matter of making the light come to a focus inside the glass so that the peak intensity at the focus spot gets higher than the damage threshold of glass (something like 10GW/cm^2). In answer to the OP's question about the "leftovers", nothing really goes anywhere. The structure of the glass just changes. Bonds are broken and new bonds are formed. Glass has many metallic impurities as well as OH- which probably make the process not so simple to analyze, but I bet it has been studied, perhaps by the guys at Livermore where they are trying to do nuclear fusion with lasers. Internal damage is visible. You can buy internal "sculptures" made this way
http://www.bathsheba.com/crystal/
A guy at work has one made from some photos of his kids. (his kids' faces in 3d inside the glass).
As far as this company's final market, it is the medical industry. The idea is to use it for surgery or something. The difference between this and previous medical lasers is that because the pulse length is short, the pulse energy required to reach high intensities is lower and the effective ablation spot size is smaller compared to longer pulse lasers.