Well, there's this red tent that the women go inside of to do girl stuff, and the narrator's a girl, so she goes inside the red tent, and so, uh, yeah.
Okay, there's a reason this book is called The Red Tent other than just the fact that there's a red tent in it. Overall, the story is about becoming a woman: we follow Dinah from adolescence to old age, and her growth as a person and as a woman is always at the forefront.
And part of becoming a woman for Dinah is her experiences inside the red tent. This is where the women gossip and shared intimate experiences. Most likely, the red tent is where Dinah heard the story of Jacob's arrival and of his marriage to her mothers. The red tent is where Dinah was brought into womanhood.
For a woman in the times of Genesis, life can be pretty scary. Back then, women had very little power. Their main function in society was to get hitched and produce heirs for their husband. Menstruation, childbirth, personal beauty, finding the right husband at the right time—all of this stuff was serious business. The red tent is the one place where these women have their own world and their own power. While they have it, it's the one place where they can be safe.
We really don't get to see the red tent in the second half of the book, because Dinah is no longer with her mothers. Nevertheless, the memory of the red tent lingers throughout the book. Dinah constantly looks back at the red tent with nostalgia and longing, and so do we.