For Nephi, this is a simple statement. He had been prophesied to become a ruler and a teacher. His people desired that he be their king, and so it fulfilled Jehovah’s prophetic promise. That answers Nephi’s purpose for including this statement, but it leaves open some very important anthropological questions.
In the study of human populations, it has been noted that there is a certain set of requirements before we have a king. One of the requirements is population. If there were no indigenous peoples expanding the remnants of Lehi’s family, there is no way that a communal structure such as a temple could have been built, and there certainly were not sufficient people for a king.
Smaller populations must spend more time making sure they are fed. Only with a larger population’s agriculture to create a stable basis could people spare the time to volunteer time to communal building projects. Only after societies have that number of people would they move from village headman to a leader that might be called a king.
Thus, one point is that this tells us that there was a reasonable sizable population. The next question is why the people wanted a king if Nephi didn’t want to be a king. That answer relies upon the archaeology of the area of the Maya culture in which they likely found themselves. This was the time period when surrounding communities were also raising up kings. Thus, it was in the air. The people wanted Nephi to be a king, not just a ruler, because kings were what other communities wanted at that time period.