Shemini Atzeret literally means “the assembly of the eighth (day)”. It occurs directly after the seven day feast of Sukkot, but it is a separate holiday and does not involve the special observances associated with Sukkot.

Leviticus 23:36

36 Seven days you shall offer an offering made by fire to the LORD: on the eighth day shall be a holy convocation to you; and you shall offer an offering made by fire to the LORD: it is a solemn assembly; and you shall do no servile work therein.

Rabbinic literature explains the holiday this way: our Creator is like a host, who invites us as visitors for a limited time, but when the time comes for us to leave, He has enjoyed Himself so much that He asks us to stay another day. Another related explanation: Sukkot is a holiday intended for all of mankind, but when Sukkot is over, the Creator invites the Jewish people to stay for an extra day, for a more intimate celebration.

It is a modest holiday just to celebrate the LORD’s special relationship with His beloved nation, Israel.