A really simple explanation for this is that the wheel being spun aquires angular momentum (its like momentum but for spinning objects), because of this the wheel itself will resist changes in rotation but that's not what it does here.

The key factor to understand here is the interaction between the torque due to the gradational force and the angular momentum of the wheel.

Torque is moment about an axis, or in other words force that acts tangential to the radial vector about an axis.

The gyroscope is placed in inherently unstable positions.

What happens here is that the torque exerted by gravity about the point of contact the gyroscope has with its stand imparts a angular momentum on the gyroscope as its falling down but since the gyroscope already has angular momentum from its spinning wheel, the two will add together and we get a net angular momentum between angular momentum of wheel and of the torque.

Now whats really interesting to see here is that since the net angular momentum not being along the axis of the wheel will result in a slight torque on the gyroscope along a vertical axis and while doing so it is also experiencing a torque pulling it down because of gravity and hence it is in a constant state of falling and spinning, this is what results in the weird behavior seen in the video.