How does Della save money?

Della saves money by squirreling away a few pennies here and there, wherever and whenever she can manage. The narrator tells us that she's

[...] saved one and two at a time by bulldozing the grocer and the vegetable man and the butcher until [her] cheeks burned with the silent imputation of parsimony that such close dealing implied.

In other words, Della has been haggling with each of the people she purchases food from throughout the week. By bargaining with each merchant, offering them slightly less than they are asking for each item, Della has been able to stash away one dollar and eighty-seven cents. However, this process makes her blush as a result of what it reveals about her family's difficult financial situation. We learn that Jim, Della's husband, used to make thirty dollars a week, but his income has since been reduced to only twenty dollars; in other words, he's lost a third of his income. We know that the Youngs' flat costs eight dollars a week, so this only leaves twelve dollars per week for food and other necessities. Della's food budget is likely already quite small, and this is possibly why she has not been able to save more.

Women had to do a lot more shopping in O. Henry's day. There were no supermarkets, and they had no refrigerators, so they went shopping practically every day. O. Henry says that Della has been trying to save money for Jim's Christmas present by haggling with the merchants she traded with:
ONE DOLLAR AND eighty-seven cents. That was all. And sixty cents of it was in pennies. Pennies saved one and two at a time by bulldozing the grocer and the vegetable man and the butcher until one's cheeks burned with the silent imputation of parsimony that such close dealing implied.
As a woman without her own source of income, Della doesn't have many options when it comes to saving money for Jim's Christmas present. One of the few things she can do is economize.
It is interesting to compare Della Young in O. Henry's story "The Gift of the Magi" with Mathilde Loisel in Guy de Maupassant's famous story "The Necklace." Mathilde was forced to economize after they went into heavy debt to replace the lost necklace. About her only way of saving money was by haggling with the merchants when she went shopping.
She came to know what heavy housework meant and the odious cares of the kitchen[...]. And dressed like a woman of the people, she went to the fruiterer, the grocer, the butcher, a basket on her arm, bargaining, meeting with impertinence, defending her miserable money, sou by sou.
With few employment options open to them, restricted by both their class and gender, Della and Madame Loisel struggle to make a difference in their own financial security, and all gains are hard-won.