With a swept wing design, a "yaw" (nose moving left or right) can "couple" or cause a rolling motion. A huge aircraft like that KC-135 tanker, rolled on it's side, no longer has the wings supporting it against gravity. It will drop like a stone, leading to excess speed, which generally tears the plane apart. They were in a lot of trouble.
Losing the 2 left engines with power still on the 2 right engines caused a yaw-roll to the left. As the plane started to sharply descend, the pilot most likely (not show in video), reduced throttle to idle. Now the 2 remaining engines caused more drag on the right side, resulting in a yaw-roll to the right.
It must be understood that any aerodynamic asymmetry at 500 knots will have very strong effects. They were rolling very hard in both directions.
Deploying the speed brakes increased drag on both wings, increasing the tendency not to yaw. We can hear (then) Major Sweeney say "I have lateral control" (can control yawing). This greatly helps ease the task of rolling the plane upright, which then allows the wing to support the plane and pull out of the dive.
Losing 2 engines may have caused the initial upset, or they could have been torn loose by the violent manuevering. We'd have to check the maintenance logs there. One account states they flew into turbulence from another aircraft.
does this trick always work?
Airbrakes (including parachutes) do tend to help straighten the flight path by increasing drag behind the center of gravity. They are also used to help recover from spins when other control inputs are inadequate.
