They are low profile since they can be folded against the body of a missile/rocket.

They also have good aerodynamic properties at hypersonic speeds.

http://en.wikipedia.org/wiki/Grid_fin for more info

* Grid fins work well at subsonic and high super sonic speeds (but poorly around Mach 1) lowering drag at high mach numbers.

* Grid fins require less structure to support / actuate them per unit control force

* Grid fins are more easily stowed (unneeded during launch, would cause significant drag forces if not stowed)

Watching their example video I'm guessing they act as steerable "fins" where the working area of the fin is spread across a number of "vanes". Not clear if there are two degrees of freedom or just one for each one.

As the rocket travels through the air each one can be rotated on its axis perpendicular to the body of the rocket. If all are rotated the same way they would impart a spin, if two were rotated one way and other two rotated the opposite direction they would simply slow the rocket down. If two adjacent ones are counter rotated the provide drag on one side (imparting a pitch change).

I was thoroughly impressed when I saw the video, that is a really cool technique.

An interesting mental model is a biplane (well, a grid of 20 not just 2) of really small leading edge slats. That's obviously not exactly how they work, but for seven word summary, not bad.

They're originally a "Russian thing", for a decade or so it seemed like every missile they designed had grid fins.

They're an example of high optimization. Around certain speeds they work REALLY well and weigh practically nothing and the control forces are very low. On the other hand they're little more than speedbrakes around other airspeeds.

Here's a Falcon 9R flight using the grid fins:

https://www.youtube.com/watch?v=DgLBIdVg3EM