This is about corruption in the government. Shakespeare uses the word State. Not country or nation, but State. He is referring to the organization. The politicians and their institutions. The idea of something being rotten is metaphor to the idea that something is no good inside of the State. If the State is supposed to work for the people, someone inside the State is not. There's a force or a person that is not working to fulfill the interest of the people. If it's rotten, it might be much worse than that. It might mean that someone might be even working against the people.
If you are trying to remember information that you are reading, you should be an "active" reader, who pays attention to the words and thinks about connections that may occur.
