It is clear that when we look at the story as a whole, the principal reason why Miss Brill is so taken and enraptured by the way in which the activity at the park and everybody there, including herself, is actually like a play, is because it allows her to believe that her life has significance and is important in spite of the reality of the emptiness of her life, as the ending of the story makes clear. Note what the text tells us about this metaphor and how it presents the play and Miss Brill herself:
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.
The play metaphor therefore allows Miss Brill to believe in two fantasies that the story show to be absolutely false. Firstly, it allows her to believe that she is significant and important. Miss Brill herself notes that she has "a part" in this drama, and imagines telling her "old invalid gentleman" that she is an "actress" to show how important she would like to believe she is. Secondly, this play metaphor allows her to feel that she is noticed and that other people take interest in her. Note how she believes that if she hadn't been there "somebody would have noticed." Of course, these fantasies are belied when she overhears the young lovers being rude about her and when she returns to her "little dark room--her room like a cupboard." The emptiness and insignificance of her life cannot be ignored even by herself.