Why did Israelite choose a king?

Answer 1

The Israelites at that time were coming to the end of an era which was unique in world history. They had undertaken a grand experiment: whether a nation could govern themselves for centuries without a king or organized government.
Instead, there were the officers of tens, fifties, hundreds and thousands (Exodus ch.18), and the court of Elders (Sanhedrin), who were Torah-scholars that provided Torah-rulings and guidance. Each of the leading Judges (Gideon, Deborah etc.) was a private citizen (not a head of government) who led the nation only during a brief episode of battle. (See: The Judges)
During that era (of the Judges, about 360 years), when someone raised the possibility of having a king, the answer was: God will rule over you, not a king (Judges 8:23). The events of the Exodus and the Giving of the Torah were so fresh in the nation's memory that they didn't need a king; God was their King. (The missteps which did take place in that era, is a subject beyond the scope of the present question.)

In Samuel's old age, over four centuries after the Exodus, the people (including their Torah-scholars) felt that the time had come to take a regular government. The Torah itself permitted this (Deuteronomy ch.17); and they saw that Samuel's sons didn't seem to have reached his spiritual level (1 Samuel 8:2-3).

Their usage of the expression "like the other nations around us" need not be a problem, since they were deliberately quoting the Torah itself (Deuteronomy 17:14).
What then was their mistake (why did Samuel criticize them)?

According to the Malbim commentary (on 1 Samuel), they should have waited at least until Samuel was too old to function.
According to the K'li Yakar commentary (on Deuteronomy), their precise choice of language ("for us," instead of the "over us" which the Torah had said), hinted that they wanted a king who might be affected by public pressure (which later happened with Rehoboam). See Talmud, Yoma 22b.
According to Samuel himself (1 Samuel 8:10-18), they were taking a regrettable risk because later kings might be overbearing.

One more point: in Judaism we have a general rule which is called "the descent of the generations." This means that according to our tradition, the earlier a generation lived, the higher was their spiritual level (Talmud, Shabbat 112b). No Talmud-sage would dare to negate a verse of the Prophets; and no later Rabbi would dare to belittle a Talmud-sage.
For this reason, we must not judge that generation. And concerning Saul, our tradition explicitly states that he was a righteous man (Talmud, Moed Katan 16b; and Midrash Breishit Rabah 54:4); and he unified the Israelites and defeated their enemies round about (1 Samuel 14:47).

See also:

More about Samuel

Jewish history timeline