The key is the pinion that drives a normal chuck's circular rack.
The keyless chuck simply makes makes the rack able to be driven by your hand. If you look at the radius of a chuck key vs a keyless chuck, they're roughly in the same ballpark.
It essentially a collet mechanism. the outer coller is threaded internally and the fingers are part of a threaded cylinder that has notches cut in it. the nose has a tapered id to drive the fingers together as the cylinder translates through the threaded collar. the three lobed shape of the nose prevents the inner finger cylinder from rotating. the keyed and keyless variety are almost identical, the only difference is that the threads are generally finer which means it takes less torque to clamp the bit in the chuck.
