From the source you provided, the answers seem to be outlined well from the authors' sentence that states:
The Pr [release probability] of SVs [synaptic vesicles] at the AZ [active zone] is set by a complex interplay of different presynaptic properties including the availability of release-ready SVs, the location of the SVs relative to the voltage-gated calcium channels (VGCCs) at the AZ, the magnitude of calcium influx upon arrival of the AP, the buffering of calcium ions as well as the identity and sensitivity of the calcium sensor. These properties are not only interconnected, but can also be regulated dynamically to match the requirements of activity patterns mediated by the synapse.
Below, I use an analogy to provide a framework for understanding how these factors come together to mediate Pr as a whole, but you can read a scientific review on that here: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3059705/. However, the actual molecular/proteinaceous machinery that underlie these phenomena are pretty well explained in your source (particularly in the sections "SV Proteins Regulating Pr" and "Recruitment of Calcium Channels to the AZ").
To use an analogy, think about the presynatpic terminal as the 100 meter dash at the Olympics. The track only has so much space, so not everyone can run the race at the same time. That's why there are multiple heats of the race. The first point from the authors is availability of release-ready SVs, which your source discusses as part of the readily releasable pool (RRP) of vesicles. Think of those SVs as the runners in a heat, lined up in the start position waiting for the gun to go off. Their equipment is ready (starting blocks set, shoes tied, proper uniform and tag on, etc.). They're all in position, awaiting the signal, and they will take off as soon as they get that signal. Therefore, the more SVs you have in that position with equipment/molecular machinery prepared to "go" in response to a signal, the more likely you are to have SV release. The second point is location of the SVs relative to VGCCs. In our example, this would be equivalent to having a lane closer to the person shooting the start pistol. The closer a runner is to that, the more likely they are to hear the pistol go off and start the race. Conversely if the person with the start pistol is at the finish line, it's possible less runners will hear it go off and miss their cue to start running. Since SVs require calcium to undergo membrane fusion and release their contents into the extracellular space, the closer these vesicles are to calcium channels the more likely they are to get that necessary substance for fusion/release. Third, the magnitude of calcium influx upon arrival of the AP, which we can think of as the volume of the start pistol. If the start pistol is loud (a high volume of calcium comes into the cell), it's more likely the runners will hear the shot go off and start running (SVs will fuse to the membrane). However, a very quiet start pistol (low influx of calcium) is less likely to be heard by the runners (less likely to bind SV proteins important for fusion). Lastly, they list buffering of calcium ions as well as the identity of the calcium sensor. What's important to remember about the process of SV fusion is that it requires free calcium that can bind to the necessary vesicle proteins, so any other proteins (calmodulin, calbindin, the list goes on forever) that bind free calcium and make it unavailable to the vesicle will decrease the likelihood of successful vesicle fusion and neurotransmitter release. This would be like someone giving the runners ear plugs before the race, which dampens the sound of the start pistol and decreases the likelihood of runners hearing their cue to start. Again, there are nuances not covered in this analogy, but I'd recommend going through your source again with this framework in mind to pick up where those nuances come in.