What he's saying: Paul has an "unhinged obsession with me," and his motive for the hearing is to "get me to say something — anything — that could vindicate his repeated public pledges that I end up, in his words, quote, behind bars unquote," Fauci said.
"Although it pains me to do so … under the advice of my attorneys, I will invoke my right under the Fifth Amendment of the Constitution to refrain from answering your questions," Fauci said.
Lawyers will almost always advise staying silent. It's genuinely a right that US citizens have, and one that's not exercised often enough. Back in February, Les Wexner was testifying to Congress. His lawyer very bluntly told him not to answer any question with more than 5 words.
The real question becomes what does Fauci have to gain or lose by answering these questions. And that's likely to be "nothing" and "everything", respectively. He simply has nothing to gain by answering questions. He (and everyone else) does not believe the goal of these questions is to get to the truth. It's to create a circus, and railroad him into a jail cell.