The Hakafot of Simchat Torah are an expression of an enclosed world that is protected by the Torah.
The Maharal explains that when you go into your Succah you are leaving one level of existence for another. Once a person has succeeded in leaving his physical world and experiencing even just for a few moments, living in a spiritual world, he will never live in this world the way he did before. He has risen to a new level. We are changed when we are able to step into a different level of existence.
This circle is not formed by us, but around us in order to enclose and protect us. The circle of the Succah creates a special space for us. The circles of Simchat Torah create a separate world for us in which we can experience an entirely different perspective on life. We can take that experience with us even when we no longer are within the circle. The circle is a cocoon.
The Midrash teaches that when the verse says, “ His left hand is underneath my head, and his right hand hugs me,” it is alluding to the Succah and Simchat Torah. The left hand is referring to the Succah and the right hand that hugs is Simchat Torah. The circles of Simchat Torah are hugs from God that provide is with a special place.
The first day of Succot is to help us pinpoint what has changed. Simchat Torah is when we remind ourselves to stop over the course of the coming year and remember that we experienced a different level of life, and to reflect on how are we different because of that experience.