About what is demonstrated with the setup in that video:
It is a setup where the rotation rate of the rotating platform is transferred to the spinning gyro wheel by the outer ring of the gimbal mounting; the orientation of the outer ring of the gimbal mounting is fixed relative to the rotating platform.
Under those circumstances: the gyro wheel spinning in the same direction as the rotating platform is a state of lower potential energy than the state of counter-rotating.
I will use the following words to refer to axes of rotation: spinning, pitching, swiveling
spinning - rotation of the gyro wheel
pitching - rotation of the inner gimbal ring
swiveling - rotation of the outer gimbal ring
Consider what would happen if the starting configuration is that the spin axis of the gyro wheel is parallel to the plane of the rotating platform. As you set the rotating platform into motion: the outer gimbal ring imparts that rotating motion to the spinning gyro wheel. In response the gyro wheel proceeds with pitching motion.
(For explanation of that pitching response: see the following discussion on physics.stackexchange, written by me, of
gyroscopic precession )
The tendency to impart pitching motion is the strongest when the spin axis is parallel to the plane of the rotating platform. At the point where the spin axis is
aligned with the rotation axis of the rotating platform there is no longer a pitching tendency.
The pitching response is quite vigorous, so the gyro wheel tends to overshoot the perpendicular orientation. Once the pitching motion has gone
passed the perpendicular: the pitching tendency arises again, acting once again to pitch the gyro wheel to
alignment with the rotation axis of the rotating platform.
In the video you see that when the angular velocity of the rotating platform is reversed the gyro wheel pitches 180 degrees. The gyro wheel was very close to being perfectly anti-aligned, and if it would actually be very, very close to being perfectly anti-aligned the anti-alignment would not immediately result in the gyro wheel pitching 180 degrees. But there is enough of an intitial angle to give a bit of leverage, and pitching motion commences.
As the gyro wheel pitches into alignment (with the rotation axis of the rotating platform) you see an oscillation that dampens out after a couple of swings. That gives an indication of the amount of friction in the gimbal bearings. That friction removes the energy that
can dissipate, and from then on the system is in the lowest potential energy state that is available. The lowest available energy state is with the spin axis of the gyro wheel is aligned with the rotation axis of the rotating platform.
Every time the angular velocity of the rotating platform is reversed the system is replenished with potential energy, which is then dissipated by way of the gyro wheel pitching to alignment.
Gyrocompass
The effect demonstrated in the video is used in a navigational device called 'Gyrocompass'. The gyro wheel of a gyro compass is suspended in such a way that the spin axis of the gyro wheel tends to go into alignment with the Earth's rotation. When the spin axis of the gyro wheel is not aligned with the Earth's rotation that rotation is transferred to the gyro wheel. Calibrated friction of motion of the gimbal suspension is optimized for the gyro wheel proceeding to alignment with the Earth's rotation.
Submarines don't have GPS reception while submerged. A gyrocompass allows the crew of a submarine to see the orientation of the submarine (with respect to the Earth).
A pure gyroscope will have some non-zero drift. By contrast: for a gyrocompass it is intrinsically impossible to have any drift. A gyrocompass in operation will intrinsically align with the Earth's axis.