The name "Shemini Atzeret" comes from the verse in Numbers (29:25): "You shall celebrate the eighth day as a time of atzeret, restriction for you."
Atzeret, the word for “restriction” can also be translated as “retaining,” implying assimilation and integration.
Shemini Atzeret is described as “a time of restriction for you,” whereas the seventh day of Passover is described as “a time of restriction for G-d, your G-d.”
This is because Passover celebrates the Exodus from Egypt, when we were not spiritually mature enough to assimilate and integrate the significance and implications of the great Divine revelations we witnessed. Therefore, these revelations remained above, “for G-d, your G-d,” kept in safekeeping, so to speak, until we were ready for them.
The holiday of Sukot, however (which culminates in Shemini Atzeret), celebrates G-d’s protection of us during our trek in the desert, most of which took place after the revelation at Mt. Sinai, after we had achieved spiritual maturity. As such, we could integrate these revelations into our consciousness, and therefore they are “for you.”
On the personal level, we are all reborn into a higher level of relationship with G-d every year. On the seventh day of Passover, a mere week after our yearly rebirth, we are not mature enough to appreciate the implications of this new awareness. Only after we have received and integrated the new Divine consciousness fully—six months later, at Shemini Atzeret—do these revelations become truly ours.
This is alluded to in this verse, “the eighth day shall be a time of restriction for you.” The imagery of the “day” connotes revelation; “eight” (in Hebrew: shemoneh) connotes “fat” or “fullness” (shemen). The verse can thus be read, “you will integrate the fullness of the revelation” on Shemini Atzeret.
This explains the explosive levels of joy and celebration of Shemini Atzeret and Simchat Torah.