Let's start with the narrative point of view. This story is told from the third-person perspective. This means that the narrator is an outside observer telling us a story about Jim and Della; however, being told from the third-person point of view isn't what makes the narrative style so engaging. What is so unique about this particular narrative is that is reads or sounds like something that is being told to us orally. O. Henry manages to do this by doing something that most writers don't do. He breaks grammar rules. People may write with correct grammar, but most people don't speak with proper grammar. When the narrator of this story blatantly ignores basic grammar conventions, it makes the story read as if someone is speaking the story to us. Take the first "sentence" of the story.
One dollar and eighty-seven cents.
It's not a sentence, because it doesn't contain a verb. It simply states a monetary amount. The narrative also starts sentences with the word "and." That's something that people might prefer to avoid in writing but do all the time in spoken conversation.
And the next day would be Christmas.
Finally, the narrative style sticks to using short sentences. This means that readers are not likely to get bogged down with big, long, convoluted sentences. This is also something that people tend to do when narrating events to each other.