The diode is biased near the break down point. Once the diode avalanches it conducts and will remain in a conducting state or have a long recovery time unless quenched. With the addition of RL there is a voltage drop when current flows which will drop the voltage far enough to terminate the conduction in some diodes. In other diodes, especially InGaAs, an additional voltage pulse is necessary to quench. Cap C is primarily a DC blocking cap removing the pulse generator from the DC model (offsets etc) and limiting the noise contributed by the pulse generator while waiting for the next photon. C can also contribute to limiting the rise time of the pulse, if the rise time is too short it may trigger a avalanche.
I will dig through my old files to see if I can find a good white paper covering component values. Recovery time, dark counts, and sensitivity are all directly affected.
I did a lot of work with APD's (Si and InGaAs) related to imaging LiDAR systems.