Inside the pump-like mechanism, there is a vertical shaft with helical grooves cut into its length, which wrap around the shaft. the bottom of this shaft is connected to the salad basket and supported by a bearing at its base, so the shaft and the basket can revolve.
The "pump" handle is a hollow tube which slips over the basket shaft. Inside it are a couple of protrusions which fit into the spiral groove. When you press down on the handle, those protrusions engage the groove and as the protrusions descend, they urge the shaft to revolve, which sets the basket into rotation.
At the bottom of the handle's stroke, the sprial groove blends into a circumferential cut or "runout groove" in the shaft so the shaft and basket can continue to revolve without any further motion on the part of the handle.
When you release pressure on the handle, a spring urges the handle back up again while the protrusions retract. At the top of the handle's travel, there is another runout groove that the protrusions fall into, and when you push down a second time on the handle, the protrusions find the spiral groove and the downward force applied by your hand on the handle forces the shaft to turn faster, and the process repeats.