Clause 1. The Senators and Representatives shall receive a Compensation for their Services, to be ascertained by Law, and paid out of the Treasury of the United States. They shall in all Cases, except Treason, Felony and Breach of the Peace, be privileged from Arrest during their Attendance at the Session of their respective Houses, and in going to and returning from the same; and for any Speech or Debate in either House, they shall not be questioned in any other Place.
Here we have another odd mash-up of two apparently unrelated points in one clause. First, congressmen get paid by the government, at a salary they set themselves. (The The 27th Amendment, proposed in 1789 and ratified 223 years later (!) in 1992, put some restrictions on congressmen's ability to give themselves pay raises.) Second, congressmen have "legislative immunity"; that is, they cannot be charged with a crime for anything they say in Congress, and they cannot be arrested or harassed by the police unless they have committed treason or other serious crimes. The idea here is to ensure that the president can't abuse his powers by arresting or jailing legislators who disagree with him.
Clause 2. No Senator or Representative shall, during the Time for which he was elected, be appointed to any civil Office under the Authority of the United States, which shall have been created, or the Emoluments whereof shall have been encreased during such time: and no Person holding any Office under the United States, shall be a Member of either House during his Continuance in Office.
People serving in office in either executive or judicial branches of the US government cannot also simultaneously serve in Congress, and vice versa. The idea here is to ensure the separation of powers between the three branches of government. Furthermore, a member of Congress can't resign from his seat in order to take another government job if that job has had its salary increased during his term. That rule is designed to prevent corruption, making it impossible for a congressman to vote in favor of a pay raise for a certain executive office, then move into that office himself.