Through the progression of Yom Kippur to Sukkoth we take all the bad from our past year and form it into a protective surrounding cloud.
Throughout Sukkoth, provided for by this external surrounding divine presence, we work daily to bring this divine illumination into us. Finally we reach a stage where it is time to remove the outer layers and expose ourselves to the new year, on the eighth day we perform the equivalent of a spiritual brith milah. We remove the rough outer protective layer (orlah) and finally begin a new year with all the energy the previous year had to offer us.
This means in each new year we bring everything with us from the year before. All the good comes of its own, and all the bad is refined to new goodness through the process of Yom Kippur, Sukkoth, and finally Shemini Atzereth.
Now that we have performed this spiritual brith milah, we are spiritually complete, ready to live a life of closeness with HaShem at every moment.
For the Jews who live outside of Israel there is an added connotation: Since the chag comprises nine days, and there are minhagim according to some communities to sit in the Sukkah on the eighth day as well, so the ninth day is likened to a birth which takes nine months. In this case the connection of the Sukkah to the womb & placenta (and likewise to the nation of Mitzrayim) is readily apparent. On the ninth and final day we leave the Sukkah and start an entirely new life, as if reborn.
[From this we can see a little the difference between living within Eretz Yisrael, and living outside. Living in Eretz Yisrael, we have a continued existence from year to year, growing from strength to strength. (Similar to a married person who experiences hitchadshut) Whereas living outside, each year we start fresh as if just born. (Similar to a single child who can't experience hitchadshut on their own)]