Miss Brill is extraordinarily lonely. She is unmarried. She teaches English students who are (presumably) the only people she talks to on a regular basis. The narrator notes that the old, invalid man Miss Brill reads to may as well be dead; this shows the complete lack of companionship and/or conversation Miss Brill gets from him. So, within the context of this story, Miss Brill has no peers (people her own age) to talk to.
One of the ways she copes with this loneliness is by going to the park (“Jardins Publiques”). Here, she lives vicariously through the lives of people she observes. She is a detached spectator and considers everyone and everything to be a theatrical play. She romanticizes everything she sees. These Sunday trips are delusional escapes from her lonely life.
Miss Brill is so lonely that she treats her fur like a living pet. She imagines the sad eyes of the fur looking at her: “how sweet it was to see them snap at her again from the red eiderdown!” She treats the fur like a companion.
When she goes to the park, she sees an old woman with an “ermine toque.” This old woman is rejected by a man and Miss Brill seems to think that the band changes their tone to reflect this woman’s feelings. This is a bit of foreshadowing, and perhaps Miss Brill subconsciously relates to this woman. When Miss Brill is insulted by the couple, she is shaken out of her romanticized weekly escape. She is painfully reminded that she is alone, much like some of the old people she sees at the park. She goes straight home to her own “cupboard” and puts the fur in its box. When the narrator adds that Miss Brill hears something crying, we are to assume that Miss Brill thinks the fur is crying. Here, Miss Brill is retreating to another delusion because she is the one crying: not the fur. The fur in its box symbolizes Miss Brill in her “room like a cupboard.”