v1-3: Samuel's sons became like Eli's sons. Perhaps Samuel was at fault for not properly restraining them, like Eli before him, and David after him. Samuel's sons were judges, not priests.
v4-6: The unfaithfulness of Samuel's sons, and the imminent death of Samuel, made the people, and especially the leaders, consider their future.
Their conclusion was "appoint a king to lead us"; but they wanted to do what the other nations did. It is a serious mistake to copy the example of unbelievers; the ungodly cannot be expected to set a godly example; political and business leaders do not generally promote godly principles; scientists and educationalists to not promote godly obedience. We are not surprised that their request displeased Samuel.
v7: In asking for a king, the people rejected God's authority over them. They should rather have looked to the Lord God to raise up leaders for them, as in Judg 2.18; 3.10, 15.
v8-9: Samuel had the task of explaining to the people the behaviour of a king, especially that he would become a burden to the people. Later, wise men suggested to Rehoboam that he be a servant to the people (1 Kings 12.7), but he rejected that advice.
v10-18: Samuel described the burden that the king would be. This is the equivalent of civil servants, military personnel, and taxation, with which a government will burden a society. In prophecy, Samuel said that, as a result of this burden, the people would cry out to God.
v19: The people were set on their decision, and would not be moved.
v20: They wanted a king, to fight their battles, and to make their decisions; they wanted to avoid personal responsibility. They were right to criticise Samuel's sons, but they refused to accept that there was another alternative.
v21-22: The people had rejected Samuel's wise counsel. The Lord God granted their request; Ps 106.15; they would have to bear the consequences of their desires.