It helps to turns the language around and split it up:

"They shall be privileged from Arrest during their Attendance at the Session of their respective Houses in all Cases, except Treason, Felony and Breach of the Peace.

[They shall be privileged from Arrest] in going to and returning from the [the Session of their respective Houses in all Cases, except Treason, Felony and Breach of the Peace].

For any Speech or Debate in either House, they shall not be questioned in any other Place."

Any Senator disclosing classified details would be violating Senate rules which would then place him or her subject to action by the Senate itself.