What does “I know people who know people” mean?

This is an idiom. If a person says this, they are getting across that in some context, they have connections to powerful people that they can use to their advantage.

You could say this idiom in a few ways. The way it is said here, they are communicating that there is an intermediate person between them and powerful people. For example, someone might say this if they have a friend whose father is the president of a company. (If you have a direct connection to a powerful person, you would just say "I know people.")

This is usually meant in a sinister way. For example, say you got in a lot of trouble at school, but your mom's friend is the principal of the school. Say your mom uses her friendship with the principal to unfairly save you from the usual punishment. Later, your schoolmate asks you how you avoided the punishment that they were sure you were going to receive. If you wanted to tell the truth but not all the details of the connections, you could just say "I know people who know people."