The idea behind detecting them relies on something in Einstein’s Relativity theory known as the Lorentz Contraction.
Einstein showed that gravitational effects are equivalent to acceleration effects, both of which are covered in his General Theory of Relativity. His Special Relativity only applied to objects moving at a constant speed and direction.
The theories basically say that a clock in motion will tick out time more slowly relative to a clock at your side. Likewise, a unit of distance in motion pass you will appear shorter than the same unit of distance at your side. The latter is the Lorentz Contraction.
Another way of looking at the LC is to imagine that you are traveling at some high rate of speed towards some destination. You can correctly view this situation as you being motionless and the length of your destination moving toward you. And so that unit of distance, via the LC, will shrink due to your high rate of speed. This of course will be in addition to the shortening due to your physically moving closer to your destination.
However, the LC for a gravitational wave, according to the video you linked to, would be in the order of a tiny fraction of the diameter of an atomic nucleus! Yet somehow they apparently were able to detect this minute disturbance in the fabric of space-time.
They actually would detect the wave in the lab by the shortening of a “measuring stick”. One measuring stick lies parallel to the direction of the wave, the other perpendicular. Only the one parallel will be effected by the passing wave and experience this extremely minor disturbance. They would compare the two ‘measuring sticks’ (units of distance) and find that they were not perfectly identical as they were before and after the wave came through, at least for a very brief instant.
...The two “measuring sticks” are actually two perpendicular tubes with light beams shooting through them and bouncing off mirrors at each end. The beams then meet at the center and combine in such away as to cancel each other out (180 degrees out of phase). Any change at all in the lengths of EITHER tube (measuring stick) will result in the beams NOT perfectly combining at the center and thereby no longer canceling each other out.