The interesting fact about the climax of "The Gift of the Magi" is that it is part of two twists in the plot. The first, is the moment when Jim comes home, sees Della and sees that her hair is cut short. That itself is not the climax, but the preamble to it. The climax comes when he explains his shocked reaction by showing her the gift that he had gotten for her: combs for her long her.
The meaning of this is that his reaction was not out of shock at her looks, but due to the fact that he now realizes that she had made a sacrifice the same way that he did. They both gave up something very dear to them for the sake of one another.
For there lay The Combs—the set of combs, side and back, that Della had worshiped long in a Broadway window. Beautiful combs, pure tortoise shell, with jeweled rims—just the shade to wear in the beautiful vanished hair. They were expensive combs, she knew, and her heart had simply craved and yearned over them without the least hope of possession.
It is after the discovery of the combs that the rest of the action begins to slow down steadily, and this is why that is the climax. After it comes the second twist: Della has also a surprise for Jim; she got him a watch chain. Although none of the gifts do any good at his point, the act of sacrifice of this young couple speaks a lot about them and reflects the love that they have for one another.