The core reading, found in Rashi (Bamidbar 29:35) and the Gemara in Sukkah 55b, takes 'kasha alai preidaschem' as God's words to Israel: after seventy bulls were offered over Sukkot corresponding to the seventy nations, on the eighth day He asks for a small, intimate feast of just one bull so that He may delight in Israel alone, like a king who, after days of grand feasting, tells his beloved servant to prepare a modest meal just for the two of them.
Rashi (Sukkah 55b) sharpens this by explaining that the seventy bulls atone for the nations regarding rainfall judged on Sukkot, while the single bull of the eighth day represents God's exclusive pleasure in Israel alone, meaning the 'separation' that is hard is specifically the parting from the nations' atonement-service and a turn to privacy with Israel.
Bamidbar Rabbah 21:24 gives a variant of the same parable in which Israel complains that the nations for whom they sacrifice seventy bulls do not reciprocate love, so God responds by telling them to now offer for themselves alone on the eighth day, reframing the 'hard parting' as God's shift from a universal feast to a private one with His own people.
An unusual expansion of the locus of separation appears in Sefat Emet (Devarim, Sukkot 19:12), who cites a midrashic parable of a king who cannot bear to part from his daughter and so asks for a small chamber to dwell among her family, applying this to Sukkot as Israel entering the shadow of the Divine Presence, so that the difficulty of parting reflects God's reluctance to leave the dwelling He shared with Israel through the sukkah, resolved through the ongoing bond of Torah study.
HaKtav VeHaKabalah (Vayikra 23:36) reads 'atzeret' itself as a term of retention rather than mere assembly, explaining that both the eighth day of Sukkot and the seventh day of Pesach are called atzeret because God detains Israel further, so that the difficult separation is not only social but experiential: the lessons and sanctity acquired over the festival are retained within a person rather than released when the festival ends.
Abarbanel (Bamidbar 29:35) situates the separation within the practical command that Israel remain in Jerusalem an additional day beyond the seven of Sukkot, and he connects the single bull and ram offered on that day, as well as the possible significance of the lambs, to themes of God's singular kingship, suggesting that the 'hard parting' is bound up with the unique character of the offerings on that day rather than being only a sentimental image.
Sukkah 47a establishes the structural basis for treating Shemini Atzeret as a festival in its own right, distinct from the seven days of Sukkot with regard to sukka, lulav, and water libation, which grounds why the parable of a separate, smaller gathering is applied specifically to this eighth day rather than being folded into the preceding seven.
Mishnah Berurah (668:6) addresses a related practical question, whether one must still sleep or eat in the sukka on Shemini Atzeret given its uncertain status as a possible extension of the seventh day of Sukkot, reflecting how the theme of separation from the sukka itself became a live halachic question among the Acharonim.
Hemshech Ayin Beis 5674 (Shemini Atzeret, 5674) offers an unusual reading in which the single bull and ram of the eighth day signify a private, almost casual gathering, 'let us see, you and I, with whatever is found,' between God and Israel after the nations have already received their due during the seven days, and ties this intimacy to the designation of the day as Simchat Torah and to the symbolism of the incense.
In a modern restatement, Rav Eliezer Melamed (Peninei Halakhah, Sukkot 7:3) explains 'atzeret' as a gathering for farewell from the entire annual festival cycle beginning with Pesach and culminating in Sukkot, framing the difficulty of separation as God's reluctance to part from His children at the close of this extended cycle, drawing again on the king-and-sons parable associated with Rashi (Vayikra 23:36).