What effect does the situation where Jim and Della exchange gifts have on the story?
The effect is to give readers an oxymoronic happy sadness all at the same time. It makes the reader sad to know that Jim has sold his watch to afford the combs for Della's hair, while knowing that Della cut her hair to afford the chain for Jim's watch. That's sad because both of their gifts are essentially useless to the other person now.
However, it is a very happy moment for the reader as well. Because the reader gets to bear witness to exactly how deep Jim and Della's love for each other is. They are each willing to give up their most valuable possession in order to get a gift for the person that they love most. It's beautiful. It's like one of those happy/sad moments that people get all teared up about at weddings. Sad to "lose" a daughter and happy that she's found true love.