The first part of section 4 placed the power of elections in the hands of the states. Allowing them to choose the time, place, and manner in which they handle the elections. This is something I believed is taken for granted today however placing the power of elections in the hands of the states allows for a major check and balance in possible election fraud. If the central government had the power to control the elections then the majority power in office would have the power to effect the outcome. On the other side if the minority power lost in another election they could claim election fraud even though there might not be any. This prevents any sign of possible fraud by either party or political group.
The second part of section 4 is actually superseded by the 20th Amendment and therefore not something we will dive into until then.
Here each House (Senate and House of Representatives) is given the authority to dictate their own rules of its proceedings, journals, elections, and punishments.
Aside from the same portion on Compensation from the Treasury which is actually covered in the 27th Amendment, this section covers the prevention of Congressional member from being arrested going to, while in session, and leaving their House. Many people are confused by this concept but we have to remind ourselves what branch of the government has the duty of enforcing the law? With that power in the hands of the executive branch, the President could use arrest as a form of intimidation to get his agenda passed or use it as a means to prevent certain elected representatives from voting on legislation.
This section also prevents members of either House from holding another Civil Office in the United States or being elected into the opposite House at the same time.