The knob pivots the rear-view mirror so that instead of the reflection of what's behind you, you see a faded reflection. This is to be used at night, so as not to be blinded by the headlights of people behind you.
How does it work? Simple: with a glass. The rear-view mirror contains not only a mirror, but also a glass, at a different angle, in front of the mirror. Try it for yourself: if you touch the bottom of the rear-view mirror, your finger is relatively close to the mirror. If you touch the top, your finger is farther away from the mirror. That's because the glass in front of the mirror is at an angle.
In the normal position, you're looking right through the glass at the mirror that reflects what's behind you.
When you use the knob, the whole rear-view mirror turns slightly upwards, so that now the mirror reflects the car ceiling (which looks dark at night) and the glass is now oriented so that it reflects what's behind you. Since it's only a glass and not a mirror, the reflection is faded. And since the mirror reflects the ceiling, which is dark or at least a uniform color, it doesn't conflict too much with the reflection on the glass.
Complicated topics explained, simply.