After exposure to the virus, it is already inside you and your immune system will start to recognize it. Is the vaccine then just a way to kickstart this process so the body can fight off the infection sooner? Is it just a way to introduce much more antigen to the system than would be present only a few hours post exposure from actual virus replication? Or is the boost coming from adjuvants?
Rabies virus enters the body, typically from a bite, and then enters nerves which it follows up to the brain. An immune response to a first exposure of a pathogen generally takes many days, perhaps weeks, to develop to the point where it's protective. This is often even slower when the pathogen is in nerves, which are relatively sheltered from the immune system, and when there's only a small amount of virus present.
After rabies exposure, people are given two treatments: They're given rabies immune globulin, which contains pre-formed antibodies against rabies, and they're also given the vaccine.
The rabies immune globulin is the most important of these. It provides immediate protection, beginning within minutes of injection. If the virus has not yet entered the nerve cell (which often takes quite a while) then this globulin will bind to and inactivate all, or almost all, of the virus.
Giving the vaccine as well is an added precaution. It will give a faster, probably much faster, response than would the natural virus. There may be only a few rabies viruses present -- it only takes one! -- and of course the natural virus is doing everything in its power to avoid making a strong immune response, whereas the vaccine is the opposite. It has large amounts of antigen, and it's optimized for making a strong immune response. (Rabies vaccine post-exposure is boosted on days 3, 7, and 14 after the first dose. That's an extremely aggressive vaccination procedure that you don't see with any routine treatments.) The vaccine might drive a protective immune response within a few days, while the natural virus might take a month, or never.
If the rabies immune globulin doesn't catch all the virus, or if it temporarily blocks it but then some escapes -- then the vaccine-induced immune response might protect them.