Shemini Atzeret literally means "assembly of the eighth." In Leviticus (Lv 23:33-43), Sukkot, or Feast of Tabernacles, is mentioned to be seven days in length with the eighth day being a "sacred assembly" where the people gather, offer sacrifices and are not allowed to do any work. This is considered a closing ceremony (Lv 23:36). In Deuteronomy (Dt 16:13-17), this eighth day is not mentioned which indicates that this eighth day has other significance than the agricultural aspects of Sukkot.
We saw in a previous post that Sukkot is prophetic to the millennial reign of the Messiah, Jesus Christ. Therefore, this eighth day is prophetic of something that is to come after - i.e., the eternal rest (the reason for no work to be done) in the eternal heaven. We also saw in a previous post that this eighth day is prophetically tied to the year of Jubilee.
It is likely because this eighth day was debated of whether it was part of the official holiday of Sukkot that it was made into a holiday of its own: Shemini Atzeret - the assembly of the eighth. It in no way detracts from its prophetic meaning but rather highlights it since it is set apart from Sukkot itself.
Simchat Torah means "rejoicing in the Torah." There are weekly Torah readings starting from Genesis chapter 1 to Deuteronomy chapter 34. On this day, the day after Shemini Atzeret, the last passage in Deuteronomy is read and then Genesis chapter 1 is read to indicate that the Torah is a circle and never ends.
This holiday also supports the scriptural teaching that Sukkot is the last feast of the Jewish calendar that was given to Israel by God: Passover starting the cycle in the first month and Sukkot ending the cycle in the seventh month (Lv 23). Other holidays have been instituted since that time but does not change this cycle.
This also helps us to understand that "Jewish New Year" or Rosh Hashanah on Tishri 1 is not the start of a new year since it is the beginning of the seventh month. This day was originally known as Feast of Trumpets. It is representative of God remembering His covenant with Israel as a nation. Therefore, the nation has be renewed in God's sight as His covenant with them still stands. This seems to be the idea as the year of Jubilee began in the seventh month (Lv 25:9) as did the beginning of a king's reign in Jerusalem. Therefore, although not specifically mentioned in scripture, these holidays do have a scriptural connection.
For additional information, see Judaism 101.
Stones to Bread
How Christians and Jews are Connected
Fall Jewish Holidays - Part 3: Sukkot