Everything was fine with Miss Brill until two people barged in on her psychological life and ruined things for her. As she is sitting in the park, listening to everyone's conversations, imagining herself to be playing a part in the play of life, as Shakespeare says in As You Like It:
All the world's a stage,
And all the men and women merely players;
two young people enter her space. She incorporates them into the idealistic reverie of her play, telling herself that she, too, has a part, and that if she did not come to the park, people would miss her. They would notice that she was not there.
The young people make fun of her, however. They make fun of the fur collar she is wearing and her looks. They cruelly say they don't know why she even bothers to show up and that she should have just stayed home.
We don't learn what Miss Brill's reaction to the comments are because instead of stopping at the bakery on her way home, as is her custom, she returns to her dark little apartment, unclasps her fur without looking at it, and puts it back in its box - like Eleanor Rigby in the old Beatles' song. She thinks that she hears "something crying." The crying is in her soul because a cruel comment shocks her into the reality that what she thought was happiness might be, instead, loneliness.
The way I see it, there are two conflicts. First, the conflict of "man vs man" - the young couple against Miss Brill because they turn her life upside down in the space of a few moments. After that, Miss Brill has an internal conflict - "man vs himself" because she is now unhappy and "crying on the inside" -- fearful, perhaps, that her life might really not have any meaning.
It is a very sad story in my opinion. It reminds me of the quote, "Where ignorance is bliss, 'tis folly to be wise" (from Thomas Gray). Miss Brill was blissful when she was ignorant of what the young couple thought of her. When she discovers what they really think, she is sad.
See the analysis here on enotes.