When you engage the self-propel, the transmission axle (blue) turns
clockwise in this view (left side of the mower):
The ratchet spring (green) pushes the ratchet key (yellow) slightly away from the axle. The ratchet key spins
inside the middle part (purple) of the pinion gear (pink) and the sharp edge of the ratchet key is pushed into in any one of the three notches (red dots), causing the the pinion gear to also turn
clockwise.
The
outer teeth of the turning pinion gear then mesh with a larger gear (not visible) on the inside of each rear wheel making it turn
counter-clockwise, propelling the mower forward.
When the engine is off or the drive clutch released, the
transmission axle does not turn, nor do the ratchet keys. So when you manually push the mower, the large gear (not visible) inside the wheel now forces the pinion gear to turn, but the curve/slope (purple) inside the pinion gear allows it to "click" over the not-moving ratchet key. This is what causes the normal clicking sound from the rear wheels.
With the engine and and drive engaged, you may also hear the clicking if manually push the mower faster than the drive speed.