In the time around that passage, each person was their own "king" so-to-speak. Instead of everyone speaking for himself, the nation wanted a king to speak for the entire nation. If all had been centered around God, they would have been speaking with one voice--God's voice. Instead what they heard was everyone voicing his own opinions, and therefore saw a need for one kingly voice.
The Israelites did not want to follow Samuel. They demanded a king! They had rejected the Lord as their king. Chapter 8 of 1 Samuel records that the Lord comforted Samuel, telling him the people had not rejected Samuel but had rejected the Lord.