Why Do Quadcopter Motors Spin Up By Themselves With Props Off?

This may be the most common question I see asked about multirotors, so clearly it's one that deserves an explanation. Why do your quadcopter motors spin faster and slower when you arm the copter with the props off? Or sometimes, two motors spin faster and two motors spin slower? This phenomenon happens in Cleanflight, Betaflight, and probably KISS and Raceflight too (although I can't confirm it).
Here's the answer.
The answer is that the PID controller is trying to compensate for the random vibrations that are coming from the motors. Since the motors don't have props on, the PID controller can't have any effect. Since it's having no effect, it keeps trying harder and harder (the motors spin faster). This creates a feedback loop which results in the motors spinning up and up.
What should you do when this happens? I'll tell you in this video.
PS: Since I know you're going to ask, the copter in the video is the Holybro Shuriken X1.