Why does the crewed version of the Soyuz rocket use grid fins on its launch escape system? It seems like extra weight to carry along during a portion of the flight.
The grid fins passively stabilize the spacecraft while the LES is firing during an abort. They pivot outward but aren't otherwise movable.
In an abort situation, the booster may be in the process of tumbling or exploding or both when the spacecraft separates. Having draggy fins toward the back of the package quickly straightens out its flight, providing the maximum separation from the booster during the powered portion of the escape, and allowing the fairing to separate cleanly from the spacecraft after the escape tower burns out.
Stable flight of a rocket requires that the center of aerodynamic pressure be behind the center of mass. The Soyuz LES abort has to carry both the orbital module (uncrewed during launch) and the reentry module (crewed) with it because the orbital module is stacked above the reentry module. The service module below that is left behind during an LES abort.
Without the fins, most of the aero pressure would manifest at the point where the base of the LES tower meets the top of the spacecraft fairing, above the orbital module. I estimate the center of mass is near the top of the orbital module when the LES is ignited, moving backwards as the LES rocket propellant is expended. With the center of mass behind the center of pressure, the stack would be unstable. Adding the fins moves the center of aerodynamic pressure back, yielding a stable configuration, much like a badminton shuttlecock.
This is in contrast to Apollo, which has no equivalent to the Soyuz orbital module, and separates just the conical command module with its LES. Because there's no fairing to separate, it's not critical to keep the capsule stable after the first few seconds of separation, and in fact the CM would normally begin to tumble after separation. Depending on the abort altitude, either a drogue parachute or canard vanes on the top of the tower would eventually stabilize it in a blunt-end-first attitude.