Texts from Shabbat discussing Shemini Atzeret
Leviticus 23:36
seven days you shall bring offerings by fire to the LORD. On the eighth day you shall observe a sacred occasion and bring an offering by fire to the LORD; it is a solemn gathering: you shall not work at your occupations.
a solemn gathering [i.e., God says to Israel,] “I have detained you [to remain] with Me.” This is analogous to a king who invited his sons to feast with him for a certain number of days, and when the time came for them to leave, he said: “My sons! Please, stay with me just one more day, [for] it is difficult for me to part with you!” [Similarly, after the seven days of Succoth, God “detains” Israel for one extra holy day.]
Take a moment to think of those time when we spend times with friends and family and then our visit/vacation is over and we must return home. We wish we could delay our trip home so that we could spend just one more day with our friends and family. God has had us close for these High Holy Days and wants to keep us close for one more day.
BT Sukkot 55b
Rabbi Elazar said: These seventy bulls that are sacrificed as additional offerings over the course of the seven days of Sukkot, to what do they correspond? They correspond to the seventy nations of the world, and are brought to atone for their sins and to hasten world peace. Why is a single bull sacrificed on the Eighth Day of Assembly? It corresponds to the singular nation, Israel.
The Gemara cites a parable about a king of flesh and blood who said to his servants: Prepare me a great feast that will last for several days. When the feast concluded, on the last day, he said to his beloved servant: Prepare me a small feast so that I can derive pleasure from you alone.
Perhaps the idea being similar to a great party with lots of people. At the conclusion of the party we turn to our closest friends and ask them to stay just a bit longer, to enjoy the night. For the first 7 days of Sukkot we needed 70 bulls, but for Shemini Atzeret, for this intimate gathering we only need one single bull.