Throughout the Yom Tov we said again and again in our tefilos והשיאנו ד' אלהינו את ברכת מועדיך.
Tiferes Shlomo (on Shmini Atzeres) comments:
והשיאנו ד' אלהינו את ברכת מועדיך כו'. כבר בארנו בפני' שהוא לשון נשואין שיהי' הקדושה שורה עמנו מזמן המועד הזה עד מועד הבע"ל ולא יפסוק כמו קדושי אשה
V'hasieinu is like the word nisuin, a marriage. When a chassan and kallah are dating or engaged, they meet, they spend time together, and then at the end of a few hours, they each go their separate ways and return to their own lives and homes. They may even decide to never meet again, and that's OK.
All that changes once the wedding happens. At the end of the wedding, once the photos are done and the band stops playing and the meal is over, the chassan doesn't go his way and the kallah her way. Once the wedding happens, the chassan and kallah are bound together and should be inseparable.
The same should be true of Y"T. It shouldn't be that you have a holiday of Sukkos, of Shmini Atzeres, of Simchas Torah, you spent some time together with the holiness of Y"T, and at the end of it all you go your way and the holiness that came down with the Y"T departs its way and that's it until next year, next Y"T, next meeting. Y"T should be like a wedding. You should leave taking the holiness of the Y"T with you and not letting it go. That's what we are asking for when we say v'hasieinu.