This was a story told to us in the beginning of widespread vaccination, but is disproven since we have evidence that vaccinated people can carry and spread the virus. The story changed to we are less likely to spread it, but there is also evidence for similar amounts of virus in vaccinated people, so I take it with a grain of salt given the history. (multiple reputable sources readily available on Google)
It's important to know and spread this information as it means that even though vaccinated people protect themselves from harsh symptoms they still have to be careful not to spread the virus.