The video is great, but it leaves out one important thing: the second solenoid, which was not shown and sits under the electric terminals, routes a small amount of pressurized gas to the bottom of the big diaphragm which effectively operates the main outlet valve (and also regulates the pressure). Gas pressure is used as the operating force for the main valve. This is important on "millivolt" versions of this valve, where a bigger 700 millivolt thermopile provides all the electricity to operate both solenoids. The thermopile would have to be very large to provide enough power to operate the main gas valve all by itself.
190055joe asked how 20 mv can operate the solenoid. The voltage does not "pull in" the solenoid, it just holds the solenoid in its fully-in position. That takes less current/voltage. The solenoid is "pulled in" when the Pilot button is pressed in by the person lighting the pilot.