She cannot be detained on her way Congress due to minor offenses. It doesn't mean that she gets away scott-free and doesn't have to answer for the offense. The idea is to prevent police from (intenitonally) keeping Congresscritters from votes for minor offenses (real or imagined), not to exempt Congresscritters from the law.
I thought it worked that a state official could make a report to congress, and congress could than act as it saw fit.
The point being to prevent a state who wanted to make a vote go a certain way from making up a crime to detain representatives of other states who had to pass through on the way to vote.
But you could see that if DC wanted to win an election on, say, voting rights, they could send their police down and round up all the republicans right before the vote, so the democrats could win for them.