Those vaccinated, because they have less symptoms, less coughing basically, might not spread it around as much, but we don't know that yet. Remember asymptomatic spreading is a big concern for this virus.
If I may pick another nit, it's not asymptomatic but presymptomatic that seem to be the major concern, from what I've heard: that is, people who spread the disease will eventually show symptoms (usually within a day or two). The balance of evidence seems to suggest that people with viral loads that never get high enough to cause illness are probably not enough to cause a significant number of transmissions, but we will have to wait for widespread vaccination to confirm it.
> it protects you from the effects of Covid-19, the disease, caused by your body's reaction to the virus.
There isn't really evidence either way for this. To quote Nature[1]:
"Tests on more than 43,000 people have shown that the Pfizer vaccine is 95% effective at preventing disease"
> vaccination doesn't mean you cannot get the virus and spread it around.
This might be true:
"But none has demonstrated that it prevents infection altogether, or reduces the spread of the virus in a population. This leaves open the chance that those who are vaccinated could remain susceptible to asymptomatic infection — and could transmit that infection to others who remain vulnerable. “In the worst-case scenario, you have people walking around feeling fine, but shedding virus everywhere"
Here's a nice video on the immune system that might help you out. Their other videos on the immune system, the complement system, and viruses might also be helpful. Not the most technical deep dives, but good, abstracted explanations.
In short, I don't think the vaccine prevents entry of the virus into the cells. That still happens, as it's a mechanical protein interaction (as far as I know). However, the vaccine (mRNA ones at least) work by producing the protein on the outside of the virus and putting that in your system, so your immune system learns how to respond. At least that's my mile-high interpretation.