I kinda suck at explaining things, and i've never taken a ASC apart, but I believe it works like a regular 3-speed Sturmey Archer hub. Basically... inside the hub shell there are mulitple gear wheels which are engaged with these gear "plates" (there are two). The gear wheels are in the middle of the two plates, and There is one large gear wheel surrounded by 3 or 4 smaller gear wheels, depending on the model hub you have. First, the cog drives one of the plates, the smaller one, and the bigger "plate" drives the hub shell. The smaller cog goes around 4 times, and the big one goes around 3 times.

The first gear (the lowest one) works like this: You pedal down and if your wheel turn 2 times, the cog is turning 3 times, thus giving you a lower gear.

The second gear is a direct drive now. The cog drives the small plate, but instead of having the large ring move the hub, the small plate does that as well. Pretty cool.

Third gear is opposite of the first gear, as the cog drives the large plate instead of the small one. So if your wheel goes around 3 times, the cog is only going around twice.

Now I think that the fixed 3-speed hub uses something to keep the cog from being able to coast while switching gears. Like a lockring and a pressure spring. So maybe when you switch gears, it puts pressure on the cog? I don't know... but I really would like to see the ASC in person.

So yea... that probably sucked, but I wanted to try. They are super ****ing cool.