I think the fur represents Miss Brill herself. She is isolated from life, just as the fur. She keeps it stored in a little box, away from everything else, just as she lives in a dingy apartment, away from everyone else. She takes the fur out, and strokes it, and calls it "little rogue" when she plans to go to the park. Stylistically, it is almost as if the author is personifying the fur. This is the only time the fur comes out of its box, and this is the only time that Miss Brill goes out into society.
As she sits on the park bench with the fur, that she calls "dear", she imagines herself a character in a play, but it is really her life playing out. She imagines that she is important, and that if she did not appear on her park bench every week, people would notice. Then the rude young couple make fun of her and her fur. It totally crushes her because their mean comments force her to face the reality of her life, which is very lonely.
When she returns to her apartment, she is downcast, and she puts the fur (herself) back in its box to be stored away. She says she thought she heard it cry because she is crying, in her heart. This again gives a sense of life to the fur, because it is supposed to represent Miss Brill's life.