Basically, what happens is that after the first pulse, the cap voltage discharges through the zener diode until it reaches the zener voltage (Vz) and stops discharging there. So on each pulse's rising edge, the "fast diode" is reverse-biased until the pulse voltage reaches the cap voltage (= Vz). At this point, the diode starts conducting, and the cap & zener together load down the output and slow/clamp the voltage past this point. The circuit doesn't affect the falling edge of the pulse, because it would involve current flowing backwards through the "fast diode". When the pulse is done, the cap has been charged to a higher voltage than Vz, but it then discharges down to Vz through the zener again, and the process repeats.
The reliance on a specific zener resistance value to do more of a "slowing" action from the cap, and less of a "clamping" action from the zener, feels a little shady, along with all the extra power that the zener must be dissipating unnecessarily during this period (personally I'd want to try adding external resistance in series with the zener, but also not familiar with the application so take that with a heaping pile of salt), but I guess it's simple and it works(?).