The RNA guide (sgRNA) is chemically caged here, so it has some molecular decorations that are removed with light. The sgRNA can't fully match the underlying DNA it's recognizing until those decorations have been removed with the laser, at which point Cas9 then cleaves the DNA strand.
Think of the sgRNA like a regex pattern. Cas9 is like the regex engine here (but instead of capturing a group, it cuts the sequence), and DNA is the search space. In this system, we essentially have a full match, and Cas9 is in position (having found the match) but it can't cleave yet, because the decorations are blocking the way. As soon as the decorations are removed by light, the sgRNA can move down into a full match, and Cas9 then cleaves.
The primary benefit here is that you can temporally synchronize Cas9. In standard systems, you introduce Cas9 into the cell, and you have no control over when it performs cleavage, since it's essentially a greedy cleavage mechanism - as soon as it finds a match, it cleaves.
This system basically introduces a semaphore, so that you introduce Cas9, allow it find a match and pause. You can then signal for cleavage to begin. As stated in the abstract, this is useful for e.g. studying the kinetics of DNA repair, since you can control when DNA damage (Cas9 cleavage) is happening.
CRISPR refers to the kind of DNA site that can be targeted for cleavage, while CAS9 is a protein that, paired with an RNA matching that site, can cleave wherever it encounters a matching sequence.
I think these guys are pairing a variation of the CRISPR approach with a laser. They pre-bond with the DNA, but the actual cleavage occurs when they shine a laser at the right frequency at the site. That lets them decide when and where on the DNA strand cleavage happens.
If anyone understands this better than me, I'd appreciate any corrections to my explanation ;P