Why does it say that he left Beis ha'Surim to rule?
Rav Sadya Gaon, Rashi #2: He came to authority like one who left jail 1 . He is poor in development of his authority. We find that Yosef became king right after leaving jail, and also David "Ani Lekachticha Min ha'Naveh me'Acher ha'Tzon" (Shmuel II, 7:8; he had no experience of authority).
Rashi: He came from a place of filth and decay 2 . The Targum of "va'Yiv'ash" (Shemos 7:21) is v'Sari.
Ibn Ezra: It is possible that the young Chacham will be king, even if he is in Beis ha'Asurim. The Aleph is omitted, like "Malefenu mi'Bahamos Aretz" (Iyov 35:11) is like "va'A'alecha Chachmah" (ibid. 33:33). He is a youth like Yosef; his Chachmah made him master. Shlomo says, do not be astounded how a king came from the opposite. Also the old king was born poor - "Arum Yatzasi mi'Beten Imi" (Iyov 1:21).
Rashbam: The "Melech Zaken u'Chsil" left the jail (womb) to be a king - after birth, he received royal honor. The young, poor Chacham also left his mother's womb; perhaps also he will get royal honor, like occurred to the former! The latter is better, for he is a Chacham, and the former is a Kesil.
Ri Kara: This explains why the young Chacham is superior. Even though he is young and poor, his Chachmah elevates him and brings him to rule. Do not say that this is when he has respite. Even when he is incarcerated in jail, his Chachmah causes him to leave and rule, like happened to Yosef!
Seforno: This is the folly of the old king. He goes out to rule from a Beis Midrash of people who veer from the company of people who debate together.
Metzudas David: The young Chachmah saved himself with his Chachmah from the jail in which he was incarcerated, like Yosef did.
Rashi: Ha'Surim is like ha'Asurim. The Aleph is omitted, like "v'Lo Yahel Sham Aravi" (Yeshayah 13:20) is like Ye'ehal.
Rashi (13) explained that the old king is the Yetzer ha'Ra, which entered man at his birth. Here he connotes that it enters at the time of his formation, from a putrid drop! However, Sanhedrin 91b concludes that it enters at the time of birth. (PF)
What is the meaning of "Ki Gam b'Malchuso Nolad Rash"?
Rashi #1: After it (R. Avigdor - the Yetzer ha'Ra) ruled, the poor one (Yetzer Tov), who is better than him, was born. He comes from Taharah, and not from filth of the womb.
Rashi #2: It is proper that he rule, for also in his kingship he veers from conducting with authority, and lowers himself among Chachamim, like the poor normally do. Also "v'Ayir Pere Adam Yivaled" (Iyov 11:12) means that he changes from being like Ayir Pere, and becomes Adam. Nolad is Na'aseh (he becomes); it is present tense.
Rashbam: Also in the kingship of the old Kesil, a poor Kesil son was born to him, like himself 1 .
Ri Kara: This explains what good a Melech Kesil loses, even though he is old. Do not say that when the nation sees that he is a Kesil, they will not appoint him to be king. Rather, even after they accepted him to be their king, he loses kingship through his folly. We find that all Yisrael came to make Rechav'am king 2 , and due to his answer "I will increase a yoke on you" (Melachim I, 12:14) he lost his kingship.
Rid: The old king, amidst kingship and wealth, becomes poor, due to his folly.
Seforno: Amidst his bundles of Mishnah and opinions [of Chachamim] that he has, he becomes poor, for he does not understand them properly, for he does not have the ways of debate to guard from erring in them.
Metzudas David: He is proper for kingship, for they say, this Ani was born for kingship. I.e. kingship is proper for him, as if he was born king.
Perhaps he explains why it is not so astounding that a poor youth will become king, and not the king's son. (PF)
They came to complain about the yoke that Shlomo had imposed on them! However, had Rechav'am heeded the elders and consented to the nation's demands, they would have made him king. Amidst his folly, he lost kingship over 10 of the Shevatim. (PF)