In a piece of literature, the exposition serves to kind of set the scene. It is meant to give the reader the background to the story. It gives them information that they need to understand the rest of the story and it sets up the conflict.
To me, the exposition of this story consists of the first six paragraphs. In this part of the story we learn that Jim and Della are poor. We learn that in some detail (how much he makes, how much she has left). Then we learn about the main conflict -- we learn that she wants to buy him a Christmas present with that paltry amount of money.
The exposition of the "The Gift of the Magi" is when the author tells the reader the background information of the story. Its is when we learn that Jim and Della are a married couple with a small amount of money. They scrape their money together to get by. It is Christmas time, and they each want to buy the other a gift. We learn that they each have a prized possession. Della has her hair and Jim has his watch.