The story is narrated as if someone were telling it to you aloud. How does O. Henry achieve this effect? Basically he breaks grammar rules. There are lots of sentences that aren't really sentences, like the opening one: "One dollar and eighty-seven cents" (1). There's no verb or action in that sentence; it just states a sum of money. We need more information about what that sum of money "means" or "does" in order to understand the sentence. We get that information in the next sentence: "That was all" (1). Although the second sentence at least has a verb, it's also technically not a complete sentence: the subject, "that," is unspecified, and only makes sense given the previous sentence.
Likewise, the narrator is fond of starting sentences with words that grammar sticklers would say you're not supposed to start with, like "And" or "Which." This also has the effect of making one sentence hinge on the sentence before. (And if you look, you'll notice that Shmoop does this sometimes too – it's part of what makes us and O. Henry sound conversational.)
Looking at those first two sentences clues us in on how the story's style tends to operate as a whole: lots of short sentences that often depend on other sentences in order to work. This technique has a way of weaving together the story across individual sentences and gives it a flow that would be broken apart by writing in more complete, self-contained sentences. It's typical of the ways we tell stories when we speak. This style keeps listeners hanging on from one sentence to the next. It also prevents them from getting lost in overly long sentences. Since when you're listening to a story you can't go back and read a sentence again, it's important that you don't get lost. If you get caught on a particular sentence it might make you lose the thread of the whole story.
Of course, as O. Henry is trying to capture that feel of telling a story orally, he also throws in plenty of addresses to his audience of listeners, as in, "Which is always a tremendous task, dear friends – a mammoth task" (20). This further creates the feeling that he is talking directly to us.