Why did he send them on the eighth day?
Malbim: Until now he celebrated for the entire nation, for the honor of the inauguration. This ended on Shemini Atzeres. From this they understood that the eighth day is a separate festival (and not part of Sukos).
Why did they bless the king?
Malbim: Shemini Atzeres is a separate festival regarding [six matters, including] Brachah, i.e. blessing the king.
What was the good for David?
Rashi and Radak, both citing Mo'ed Katan 9a: When Shlomo wanted to bring the Aron into the Devir. The gates clung to each other. He said "Se'u She'arim Rosheichem" (Tehilim 24:7); they did not open. When he said "Zachrah l'Chasdei David" (Divrei ha'Yamim II, 6:42), immediately he was answered, and all Yisrael know that Hashem pardoned David's sin.
Radak: He fulfilled His promise to David, that his son will build the Beis ha'Mikdash.
What was the good for Yisrael?
Rashi citing Mo'ed Katan 9a: Their sin of eating on Yom Kipur was pardoned. A Bas Kol (voice from Heaven) announced that they are summonsed for the world to come.
Radak: Yisrael had Shalom and serenity in Shlomo's days. All kings served Shlomo and brought gifts to him.
QUESTIONS ON RASHI
Rashi writes that seven days were for Chanukah, and seven days were Sukos. If so, "[he sent the nation on] the eighth day" was Shemini Atzeres, Tishrei 22. Divrei ha'Yamim II, 7:10 says that he sent them on the 23rd!
Radak citing Mo'ed Katan 9a: On the eighth day they received permission to leave. They did not travel that day, for it was Yom Tov. On the 23rd they received permission again, and went to their tents.
Malbim: They could not leave [even on] the night after Yom Tov, for Shemini Atzeres obligates Linah (spending the night in Yerushalayim].
