Complex sentence: It directly follows the Jewish festival of Sukkot which is celebrated for "seven" days, and thus Shemini Atzeret is literally the "eighth" day; it is a separate--yet connected--holy day devoted to the spiritual aspects of the festival of Sukkot.
Split into simpler sentences: It directly follows the Jewish festival of Sukkot which is celebrated for "seven" days, and thus Shemini Atzeret is literally the "eighth" day. It is a separate--yet connected--holy day devoted to the spiritual aspects of the festival of Sukkot.