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Shemini Atzeret is a very unique Yom Tov. The word Atzeret comes from the Shoresh Atzar, meaning to stop and refrain. The only Mitzvah on this day is to refrain from Melacha. This is unlike every other Yom Tov, which has its own special Mitzvah whether it is connected to the Beit HaMikdash or is even practiced B’zman HaZeh. Hashem tells Klal Yisrael, “Kasha Alai Peridatchem”, “Your departure is difficult for Me” (Rashi, Bamidbar 29:35). We went through Elul, Aseret Yemei Teshuvah, and Sukkot, but that is not enough. Hashem wants one more day with Klal Yisrael. No special mitzvot to fulfill, just a day to connect to Hashem. Rav Menachem Azariah MiPano points out that Shemini Atzeret is never called a Chag in the Torah. The word Chag is related to the word Chug, meaning a circle. The Yamim Tovim that are called Chag are part of a larger cycle, while Shemini Atzeret stands at the center of that circle. All of the Yamim Tovim surround it.This is because the Kirvat Elokim which we have on Shemini Atzeret is on a higher level than any other Chag. Since the Dveikut is higher on Shemini Atzeret, it is the center point of all the Yamim Tovim. Based on this, the Chatam Sofer explains that in general, the Mitzvah of Simchat Yom Tov only applies during the day because the main Mitzvah of the Yom Tov is usually done during the day. For example, we shake Lulav and Etrog during the day. When it comes to Shemini Atzeret, the Mitzvah of Simchah even applies at night because Shemini Atzeret is not connected to any Mitzvah. Hashem wanted to spend one more day with us, and that is the holiest and highest level of a Yom Tov we can get. The Chatam Sofer goes so far as to say that the Kedusha of Shemini Atzeret is even greater than the Kedusha of Yom Kippur.
Once we are on the topic of circles, there is another type of circle which takes center stage on Shemini Atzeret/Simchat Torah, which is Hakafot. Rav Schachter explains why the Gra did not want anyone standing in the middle of the circle during Hakafot. Hakafot are modeled after the Gemara in Taanit (31a) which speaks about how Tzadikim in Gan Eden will all be in a circle equidistant from Hashem, who will be in the middle. Throughout Sukkot we are doing Hakafot and walking around the Sefer Torah, which expresses the fact that through us doing Mitzvot we achieved the ultimate goal of Limud HaTorah. Then, on Simchat Torah, Hakafot take on a different form. We dance around with the Torah with Hashem in the middle, and we are all equidistant from the center of the circle, just like how the Gemara in Taanit portrays Gan Eden for the Tzadikim. This is why the Gra did not want anyone in the middle, because Hashem is in the middle. We are showing that through Limud HaTorah we can achieve a Dveikut to Hashem. Just as Shemini Atzeret is the spiritual center around which all the Yamim Tovim revolve, so too Hashem is the center around whom we dance with the Torah. In both circles, the message is the same: every Mitzvah, every Yom Tov, and ultimately all of Torah is directed toward one goal. As David HaMelech says, “VaAni Kirvat Elokim Li Tov,” “But as for me, closeness to Elokim is good” (Tehillim 73:28)." The ultimate purpose of Torah and Mitzvot is not simply to complete a cycle of religious obligations. It is to bring us closer to Hashem. That is the message of Shemini Atzeret: after everything we have done, Hashem wants us to stop, remain with Him, and recognize that the greatest gift is the opportunity to be close to Him.