Restrain the festival by bonds to the corners of the altar (Psalms 118:27).
The Talmud states that if a person celebrates the day after the holiday with a festive meal, it is considered as though he had built an altar and had brought sacrificial offerings upon it (Succah 45b).
Rashi states that the reason for the eighth day, Shemini Atzeres, can be explained with the parable of a king who invited his children for several days of feasting. When the time came for them to leave, the king said, "Your departure is so difficult for me. Please stay with me for yet one more day" (Rashi, Leviticus 23:36). Similarly, after seven days of Succos, in His great love for Israel, God asks us to stay with Him for yet one more day before returning to our mundane activities, which so often distract us from Him.
To indicate that we cherish our closeness to God just as He does, we add a day of festivity after the last day of the holiday, to extend even further the intimate companionship with God. This testimony, that we value our intimacy with Him and that we leave the Sanctuary only because we must tend to our obligations, is held equivalent to building an altar and bringing votive offerings.
Indeed, God wants us to engage in work - Six days shall you work (Exodus 20:9) - but our attitude toward the workweek should be that of a person who is away from home on an assigned duty, and who longs to return home to his loved ones. The importance of our closeness to God should be manifest not only on the day following the festival but all year round as well.