Chasam Sofer observes that in the leining for the first days of Y"T we have what sounds like two separate descriptions of Sukkos:
1) "Dabeir el Bnei Yisrael leimor bachamisha asar yom lachodesh ha'shevi'i ha'zeh chag hasukkot shivas yamim l'Hashem." (23:34)
2) "Ach b'chamisha asar lachodesh ha'shevi'i b'aspichem es tevu'as ha'aretz tachogu es chag Hashem shivas yamim, bayom harishon shabason u'bayom ha'shemini shabason." (23:39)
From 23:34-38 the Torah only discusses the korbanos of the chag and there is no mention of the eighth day. In the second section, 23:34-43, the mitzvos of lulav and sukkah are mentioned, and right in the introductory pasuk we are told that there is an extra day of "yom ha'shemini."
There is a dual identity to the chag of Sukkos. On the one hand, it marks the culmination of the y'mei hadin of Rosh haShana-Yom Kippur-Sukkos, ending in Hosha'na Rabbah. On the other hand,it is the final one of the shalosh regalim of Pesach, Shavuos, and Sukkos.
The y'mei ha'din are a spiritual cycle of time. The shalosh regalim are essentially part of the agricultural cycle of planting (Pesach), ripening (Shavuos), and harvesting (Sukkos).
In the first section, the Torah deals only with the chag's spiritual dimension: korbanos. There is no mention of it being the harvest season, unlike in the second section, which begins by mentioning that we celebrate "b'aspichem es tevu'as ha'aretz." There, in the second section, the Torah discusses taking the lulav, celebrating nature, which has once again delivered it's bounty. There we have the command to sit in sukkah to remind the farmer not to become spoiled and self-indulgent. And there we have an added "yom shemini" tacked on to bring the farmer back to a focus on the spiritual, to have one final day devoted to ruchniyus, as a culmination of the chag.