Shemini Atzeret is essentially the eighth day of Sukkot. The literal translation is eighth day assembly. Regarding the word, assembly, according to commentary, this has to do with the connotation of the pilgrims from outside of Jerusalem, remaining behind after the Sukkot celebrations, for one more day, to rededicate oneself to to G-d’s service, imbibing the teachings from scripture, (G-d’s Word), and staying in the Temple area before going back to the daily grind (paraphrase of Sforno’s commentary).
Moreover, let it be understood, that during the seven days of Sukkot, there are 70 bulls offered for the seventy nations of the world, connecting the first seven days of Sukkot with the gentiles. Yet, Shemini Atzeret, the eighth day, is a day of assembly, in specific, solely for the Jewish people, as if H’Shem would like the pilgrims to remain in Israel for an intimate time of connection with G-d.
Regarding the pasuk, “On the eighth day there shall be an assembly for you” (Numbers 29:35), the Sfas Emes conveys an insight, that “it is for you because the gates of teshuva are open to all. But Israel takes greater joy in accepting G-d’s service anew than they did in having their sins forgiven” (p.372, The Language of Truth). Therefore, it can be said, that while the focus of Rosh Hashannah was on repentance, and the Day of Yom Kippur on forgiveness, Shemini Atzeret, a holiday connected to Simchas Torah, has a focus on renewal – the natural complement of a complete teshuvah.
This makes perfect sense, following the “shedding of sins,” as symbolized by beating the aravah (willow leaves), at the end of shachris (morning service) on Hoshannah Rabbah, the seventh day of Sukkot. After this final release of the previous year’s sins, a feeling of renewal is definitely appropriate, if everything was “done right,” in regard to teshuvah (repentance). Like, “the cleansing of the soul,” in preparation for a new year of service to G-d, via the spiritual growth, and perfection of character that result from selfless dedication to the higher values of Torah.
Ultimately, renewal may be said to involve purification through a rededication in one’s life to the service of H’Shem. This dedication may be exemplified, as is found in Bereishis, “And G-d took the man, and put him into the garden of Eden to av’dah it and to sham’rah it” (Genesis 2:15). The root of avdah, AVD (ayin–veis–dalet) connotes avad (to serve), while shamrah, SMR (shin–mem–resh) connotes shamar (to guard).
Traditionally, these refer to serving G-d through the positive commandments, and guarding ourselves against the negative commandments. In summary, our avodah (service towards G-d), and observance of the commandments. So, when we start the Torah cycle anew, we read in Bereishis about the beginning of creation, and are reminded of the main purpose of life, our avodah, overall service towards G-d, and our shomer, otherwise understood as the guarding of our souls from all that would taint the holy neshamah.