Fauci's more detailed quote from your link:

> So even though there are breakthrough infections with vaccinated people, almost always the people are asymptomatic and the level of virus is so low it makes it extremely unlikely — not impossible but very, very low likelihood — that they’re going to transmit it

And my understanding is that, in May of 2021, that was very much true and that studies did show that the vaccine greatly reduced transmissibility. With new variants, though, the original vaccines, while still being protective against severe disease, did a much worse job against preventing mild illness and transmission.