The Midrash likens this to a king who holds a seven-day celebration for his sons. On the eighth day, when it comes time for them to leave, he is reluctant to see them go and asks them to remain for one more day of celebration.
There is something special about this holiday, Shemini Atzeret, that actually prevents the departure from taking place at all.
This concept is reflected in the precise language of the Midrash. "Your departure is difficult," the king tells his sons, not "our departure."
This alludes to the fact that G-d never abandons the Jewish people; His love for us is constant and eternal. "Your departure is difficult," G-d tells us. G-d doesn't want us to abandon Him; He therefore requests that we celebrate one more holiday together which will serve to strengthen our bond.
The entire theme of Sukot is Jewish unity. When we are united with one another our relationship with G-d is strong. But, AFTER Sukkot there may exist the possibility that we will revert to our self-centeredness.
And so in order to prevent this from happening, G-d asks us to remain with Him a while longer; to celebrate a holiday which will secure our unity in an everlasting manner.