That's a fairly week post-hoc claim that was oversold in the media. The actual clinical, controlled studies weren't checking for stopping transmission.

I admit I don't know the narrative in the US/CDC. For what I understood reading around on various international news networks the idea it would prevent transmission via reducing the viral charge to almost zero was related to the variant to which the current version of the vaccines were built for. The new variants (delta first and omicron in particular) require new updated and targeted vaccines to keep the viral charge so low to keep avoiding transmission.

But again, I am no virologist and my understanding is just a sum of what I read around.