The main reason why Miss Brill enjoys her Sundays in the park is because this type of distraction is the sole medium by which the expatriate English teacher, who lives alone, has a chance to feel as a part of a bigger society. Aside from her Sundays at the park, she leads an otherwise dull existence, and a very lonely one, at that.
When Miss Brill goes to the park, she does not just go to rest and enjoy the view. She has made it a habit to people watch. More than that, she actually assigns roles to the people whom she watches. Doing this, she feels as if she knows them, and that she can approach them. This is something she would not be able to do otherwise.
And then she too[...]and the others [...]would come in with a kind of accompaniment–[...]. 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[...]
The thing is that Miss Brill has taken this habit too much to heart. She sees the people in the park as actors in a huge play. She sees herself as part of the scenery, and this is what makes her so emotional about the whole thing: she finally gets to be a part of the world, in her terms.
So strong is her attachment to the fantasy, that she does not ever realize how strange she looks like in the eyes of the world, or how odd the whole thing is. We are never told, as readers, whether Miss Brill even acknowledges herself as it is; whether she recognizes her awkwardness. So detached is she, that she even "hears something crying" when she has to rush home after hearing the young couple laugh at her and her "old smug". Especially, when they laughed at her beloved fur.
Therefore, the Sunday trips are mental escapades for Miss Brill. In them, she gets to be whoever she wants, and feel freely accepted by the rest of the world around her. We are hinted that she is, perhaps, taking this habit too seriously, and becoming too attached to it. The crying at the end shows us that her fantasy has finally cost her a fragment of her pride and dignity, after all.