The Cas9 protein (from the CRISPR system) has 2 functions:
1) It binds to DNA based on where a template RNA strand matches DNA in a genome. That RNA strand has 2 pieces - a piece that attaches to the Cas9, and a piece that matches whatever sequence of DNA you want to bind.
2) It cuts the DNA it has bound to.
(The first property is very special to Cas9. The second property is actually pretty common. The common intention in using Cas9 is to cut DNA at a place that needs editing and hope the cell's own repair machinery fix it properly.)
Important to understand about biology/chemistry, things rarely happen "IFF and only IF", but more often, happen at a rate (i.e. 1 event per microsecond).
So better more true functions of Cas9 are actually:
1) Cas9 spends a larger fraction of time near its target sequence than it does around random sequences.
2) Cas9 cuts DNA ever 1/N timeperiods, and that might be sped up when it is attached to DNA.
So Cas9 should hang out around its target sequence, most of the time, and that the cutting event happens relatively randomly. So _in general_ Cas9 will cut at its site of interest, however, it will also cut elsewhere, purely statistically speaking. And if you want to be guaranteed a cut at your spot of interest, you're likely to get cuts elsewhere ("off-target" cuts to DNA. These are undesired as you really don't want random changes to your DNA that you're not asking for...).
----- now comes the paper -----
If you make it so Cas9 cannot cut because the homing device is kinda kinked or boxy or otherwise not quite right (but it is still able to home in on particular sequences of DNA), you can make it so Cas9 runs around and finds where its supposed to be, but just hangs out there; Function 1 still works, but function 2 is turned off. These scientists made a special version of RNA that only permits the Cas9 to work when light is shined. So the RNA absorbs the light from a laser, breaks a seal, and the Cas9 snips right then and only then (in a matter of seconds). And any Cas9 molecules that did not get light cannot cut - they just continue to hang out and don't do any damage that they otherwise might do, statistically.
If you're trying to cut a specific sequence and only a specific sequence, this increases the statistical chance you are correct by shortening the time period (and spatial localization) when Cas9 can actually take its second, cutting action.