"Privileged from arrest" is the key concept. The idea is that a duly elected member cannot be prevented from voting and speaking unless he's charged with one of these crimes.
There is no protection from legitimate warrants for a congressman's office. If there were, we'd have created a law-free zone right on Capitol Hill. If that's what Hastert thinks the Constitution requires, he's too dumb, constitutionally and politically, to serve in any major public office, let alone be our leader in the House.
No, "privileged from intimidation" is the key concept.
The Executive Branch has obtained and examined Constitutionally privileged legislative papers.
The evidence they were looking for was not privileged of course, but they had to look through the privileged documents that evidence was hidden in to find it. They took minimalization procedures, but that is what happened.
"[T]he "central role" of the Clause is to "prevent intimidation of legislators by the Executive and accountability before a possibly hostile judiciary . . ."