Okay, let's take a break from all the subtlety. The room is right in the title of the book, so we know it's important. And at the end of the book she puts its meaning right out there. We're going to set it out so you know how important it is:
a lock on the door means the power to think for oneself. (6.12)
We all know this, right? Think about being a teenager (whether that was 20 years ago or, you know, right now). The ability to close your door and do your teenage thing, whatever it is—write poetry, play your guitar, make youtube videos, look up pictures of One Direction—is super important to figuring out who you are and what you want out of life. It means you have a few minutes without your parents nagging you to be who they want you to be and decide what your very own life's work is going to be.
Well, for a long time, women were basically treated like teenagers who were never allowed to lock the door. For Woolf, having a private room means a female writer can get out of the drawing room where others are always interrupting her.
For Woolf, the problem with writing in the living room is that it forces women into writing books about people's relationships with one another, and a locked room offers a woman the privacy to concentrate on the individual's relationship with important things like the universe or truth.
We have to ask, though: what's more real or more important that our relationships with each other? Isn't Woolf being a little sexist here herself, by saying that things that women write about—relationships and people—aren't as important as abstract concepts?