Clearly, the truth of this statement is shown in the way that Bella and Jim both sacrifice their most treasured possessions to buy a gift for the other. In spite of their poverty, we are told that this couple have two objects of inestimable value:
One was Jim's gold watch that had been his father's and his grandfather's. The other was Della's hair. Had the Queen of Sheba lived in the flat across the air shaft, Della would have let her hair hang out the window some day to dry just to deprecate Her Majesty's jewels and gifts.
It is these two objects that both sacrifice as a symbol of love for the other, so that they can gain the money they need to buy the perfect present for the other. Note the cost of Bella's sacrifice when she has her hair cut off to be sold and goes back home to repair the damage:
She got out her curling irons and lighted the gas and went to work repairing the ravages made by generosity added to love. Which is always a tremendous task, dear friends--a mammoth task.
The narrator signals to us the massive sacrifice she has made by describing her efforts to do something with her hair and to neaten it, as Bella realises the full extent of what she has done. And yet, at the end of the story, the narrator is clear that Jim and Bella, through their actions, captured the original spirit of giving gifts started by the Magi. Love, it seems, involves sacrifice.