As Miss Brill observes the park-goers sitting near her and walking by, she makes an allusion (perhaps inadvertent) to Shakespeare's comedy "As You Like It:"
"They were all on the stage. They weren’t only the audience, not only looking on; they were acting. Even she had a part and came every Sunday. No doubt somebody would have noticed if she hadn’t been there; she was part of the performance after all."
In a monologue from "As You Like It," a character states that "All the world’s a stage / And all the men and women merely players." In the centuries since Shakespeare wrote the play, these lines have become some of his most famous. They're often quoted in a positive light, to assert that individual lives — however humble or quotidian — are just as important as the dramatic events depicted in plays. Yet the monologue as a whole is quite pessimistic, arguing that however unique someone might seem, all people are on the same trajectory toward old age and the "mere oblivion" of death. Ultimately, the monologue argues that individual lives are both predictable and sordid.
Miss Brill may not be aware of the allusion she's making when she draws a comparison between the people in the park and the actors in a play. She certainly doesn't reference "As You Like It" directly. But her use of stage imagery reflects the contrast between the monologue's actual content and its existence in the public imagination. Miss Brill uses the comparison to put a positive spin on her solitude, suggesting that her lack of family and friends doesn't matter because all the park-goers are participants in a human drama that connects them all. Her conception of the stage recalls the popular use of these lines from Shakespeare. But by the time Miss Brill returns to her apartment at the end of the story, the cruel insults of the young couple have dispelled her notions of connection to a broader community. Instead, she must face the reality that she's an impoverished woman approaching old age alone. Here, it's clear that Miss Brill's life is proceeding along the same grim path Shakespeare describes in his monologue.