The story opens and closes with scenes involving Miss Brill's fur. What is the significance of the fur?

Miss Brill is a lonely spinster who goes to the park every Sunday to listen to the band playing and to watch the people in the area. Her own life is quiet and rather empty, so she derives a great deal of happiness from observing other people's lives, and imagines herself as a silent participant in the activities and conversations that go on around her.

Miss Brill always looked forward to the conversation. She had become really quite expert, she thought, at listening as though she didn't listen, at sitting in other people's lives just for a minute while they talked round her.

On this particular Sunday, the weather is cool enough to warrant extra clothing, so Miss Brill wears her fox-fur necklet. Fur has always been a luxury item, and it is evident that Miss Brill has had this particular fur for a long time—long enough to have become emotionally attached to the fox, and to think of it almost as a kind of pet:

Little rogue! Yes, she really felt like that about it. Little rogue biting its tail just by her left ear. She could have taken it off and laid it on her lap and stroked it.

Wearing the fur makes Miss Brill feel dressed-up, but that is appropriate, because this is the first Sunday of the "Season" and many people are out and about wearing their best clothes.

There were a number of people out this afternoon, far more than last Sunday. And the band sounded louder and gayer. That was because the Season had begun . . . Wasn't the conductor wearing a new coat, too? She was sure it was new.

Miss Brill listens in on the conversations happening around her and watches the people come and go, imagining their little dramas and romances and mentally setting them all to the music the band is playing. She muses that it is like being in the audience at a theater:

Oh, how fascinating it was! How she enjoyed it! How she loved sitting here, watching it all! It was like a play. It was exactly like a play.

It suddenly occurs to Miss Brill that she herself is part of the play rather than a simple spectator. She has an assigned role in the weekly ritual of Sundays in the park, and she is delighted to think that perhaps other people look forward to her "performances." She has the wonderful feeling of being included in something large and important, and the band seems to be playing especially for her:

[What the band] played was warm, sunny, yet there was just a faint chill—a something, what was it?—not sadness—no, not sadness—a something that made you want to sing. The tune lifted, lifted, the light shone; and it seemed to Miss Brill that in another moment all of them, all the whole company, would begin singing. . . . And Miss Brill's eyes filled with tears and she looked smiling at all the other members of the company. Yes, we understand, we understand, she thought–though what they understood she didn't know.

Into Miss Brill's beautiful reverie two new voices come, a pair of young lovers, and Miss Brill, deeply moved by her feeling of belonging to the scene all around her, listens to what they are saying. She is mortified to hear the boy describe her as "that stupid old thing," but it is the girl mocking Miss Brill's fox-fur which is particularly heartbreaking. The fur, Miss Brill's only luxury, is reduced by a few careless words to something ugly, outdated, and embarrassing. The boy implies the same is true of Miss Brill herself:

"Why does she come here at all—who wants her? Why doesn't she keep her silly old mug at home?"

Miss Brill's enjoyment of her Sunday ritual evaporates. When she gets home, she removes the fur and casts it swiftly into its box, trying not to look at it. This is the same fur which, at the beginning, she so carefully brushed and cleaned, and proudly wore when she stepped out, and thought of as a "dear little rogue." Now, seared by embarrassment, she can't bring herself to remember how much she valued it. As she puts the lid on the box, she imagines she hears "something crying," as if the fox itself is wounded by what has just happened.
The fur is just an accessory, but in a way it represents Miss Brill's self-worth. She lives a humdrum, lonely life and only connects with people obliquely, by watching and listening to them. On the day she wears the fur, however, she feels a greater and deeper connection to the people around her, and even briefly transcends her ideas of herself when she realizes she is an "actress" on the small stage of "Sundays at the park." She feels good and worthy and important—all feelings which are destroyed by the cruel remarks of the teenagers sitting nearby. The fur made Miss Brill fee dressed-up when she came to the park. On her way home, she feels ridiculous. All her sense of connection to the wider world has been shattered, and by shamefacedly putting the fur away, Miss Brill seems to be giving up on ever re-establishing that connection.