The last day of Succot that is known as שמיני עצרת and שמחת תורה, is considered a חג בפני עצמו, a holiday by itself. For this reason, we make the Bracha of שהחיינו for candle lighting and for Kiddush.
The Rabbis explain that Hashem initiated this holiday when He said the words, קשה עלי פרידתכם, it is difficult for Me to say good by to you. Hashem realized that the Jewish people were going back to their respective homes to begin the long, cold, winter. It would not be until the spring with the holiday of Pesach that all would again appear at the Beit Hamikdash.
The holidays of Rosh Hashana, Yom Kippur, and Succot, brought the people to a very high level of closeness to Hashem. It would have been a shame to have to end this connection so abruptly. Therefore, we were asked to stay another day and שמיני עצרת became its own holiday.