Ok, here is the quick and dirty: VIRAL vaccines (as opposed to bacterial vaccines) are designed to prevent SEVERE ILLNESS AND DEATH, and they do NOT always prevent viral entry and subsequent viral replication in your cells. For example, the flu shot does not prevent you from contracting flu and getting sick, but it is mostly effective at preventing severe illness, hospitalizations, and deaths.
The long, complex explanation:
Viruses are not "alive." They are microscopic particles consisting of DNA surrounded by a shell of sorts. They are similar to parasites. They cannot "live" without a host. So, when you are infected by a virus, that means the virus organism has entered YOUR cells, hijacked them and turned your cells into little replication factories that churn out more and more virus infected cells. When your cells are so full of virus DNA, they explode and release new virus particles, which then hijacks new healthy cells to start the process again. This cellular death is what triggers your immune system to respond. It goes "what caused all those cells to die?" Once your immune response messenger cells determine what is killing all the healthy cells, it identifies the pathogen and starts attacking it using antibodies and other killer cell types. At this point, you are heavily infected, and your symptoms are the result of this cellular warfare and the side effects of all the destruction.
When you get a vaccine, it sort of simulates this process, but with weakened or inactivated virus, as like a practice run. Your immune response learns how to fight the virus and then remembers how for next time. It is ready. This is why sometimes vaccines make you feel a bit under the weather.
Now imagine you are exposed to the virus later. Your immune response begins when the first cell that has been infected dies off. The response is fast and furious, and your immune system quickly takes care of the virus, well before it has replicated out of control. You don't get sick, because the amount of cell death was minimized and contained. However, you DID still get infected, technically.
Now, these vaccines are designed to detect the protein spikes that cover the surface the SARS-COV2 virus. The hope is that by targeting this spike, they can train your immune system to detect the presence of this spike as an enemy. The virus uses this spike to attach to the ACE2 receptors on the surface of cells in our noses, throats, and lungs. The spike is kind of like a key and the ACE2 receptor is like a lock. The key has to fit in the lock, which is what allows the virus to inject its DNA into our cells. Ideally, this vaccine teaches our cells that this "key" is an enemy, so our cells should NOT allow it to attach to the "lock." In essence, we are trying to get our cells to block the key, so that the virus CANNOT enter our cells, and instead simply gets eaten by our innate immune cells. The goal is to prevent viral replication in theory, but in practice, it obviously did not work 100%.
In the case of this particular vaccine, 6% of vaccine recipients contracted the virus and then showed symptoms, which is the definition of "Covid-19." Covid-19 is the disease caused by SARS-COV2.
Someone who is asymptomatic, but tests positive for the presence of the virus does not "have Covid-19." Essentially, this means the person has an excellent immune system that quickly neutralized the virus before it spread to lung cells, or in the case of the vaccine trial, the vaccine worked. It is unknown how many vaccine recipients tested positive for the virus while remaining asymptomatic. If that data was collected during the trials, it may be part of the data set that will be reviewed, although at this time, only one vaccine company has publicly stated that they DID collect this data during the trial and it DID indicate the vaccine prevented INFECTIVITY (which means that no virus particles capable of replication were detected during routine nasal swabs). That is good news.