Drivers have individual preference as to whether they use the clutch or not. Jim Holtom uses left foot braking, which means the clutch is used only from a standstill. The transmission has no synchromesh and uses straight-cut gears for maximum strength. This means that upshifts are accomplished by momentarily lifting the throttle and sliding the gearshift into the next gear. It's very fast.
Downshifts are more complex.
The gearbox is placed in neutral when the throttle is lifted for braking. Then the lower gear is selected. If the driver matches the speed perfectly, the gear engages like magic. If not, the gears are ground. If the gear speed is completely mismatched, the rear wheels may momentarily lose traction, causing the car to spin.