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.
On the second video they show them gimbaling towards the rocket and down as well as rotating so I think they have at least 2 degrees of freedom to aide in control alongside the engine gimbal.
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.
On a completely unrelated side note; the cattle grazing (or running?) in the foreground was quite a contrast to the modern, huge rocket landing itself in the background.
