What does the woman behind the wallpaper represent? How does the narrator come to identify with her?

The woman behind the wallpaper represents the narrator herself, which is why she comes to identify with the woman. Over the course of the story, the narrator gradually sees this woman in more detail because as she descends further into madness, she also becomes more and more aware of her oppression. 
The narrator is being treated for postpartum depression by "the rest cure," which means she must be isolated and not permitted to work. She must "rest" and not overexert herself through writing or socializing. The woman's husband is her doctor, and he takes her to an old house to recuperate. Male characters apparently do not understand how to treat this "female problem," and the cure clearly does not work. The narrator does not get better but rather becomes increasingly insane. However, her madness gives her insight into her position in a patriarchal society. 
The narrator first hates the wallpaper, but then she becomes interested in it, probably because she has no real outlet for her creativity. She eventually sees a woman, and later, she sees the woman moving. She describes the woman as crawling around quickly behind the paper and shaking it. She recognizes that the woman is trapped behind the paper. She also thinks there are "a great many women behind" the paper, which indicates that the woman's oppression is not just her own but is a larger societal issue. At the end of the story, the narrator is crawling around the room and she even says "I've got out at last." This shows that the woman has so fully identified with the woman behind the wallpaper that she considers them one person.