Samuel effectively led them through some close scrapes and difficult days, but eventually he was old. His sons could not be depended upon for leadership. The answer seemed obvious to the leaders of Israel. The cure-all for their difficulties, as they saw them, was a king. In human terms, this made sense.
Kings pull things together.
Kings raise armies and protect the national interest from enemies. Kings make treaties with neighbors. Kings give a sense of direction to policy, a sense of identity and dignity to the nation, and a sense of cohesion to heal divisions within the body politic. Yes, a king seemed the obvious answer to Israel's problems. What Israel forgot, and her history could have told her, was another and better answer to her problems.