The entire story is written as though the author is talking directly to the reader. Some parts of the narrative are more directly worded to reflect this approach, but the whole story is written in a very conversational tone.
In the third paragraph, the reader is told to look about the apartment while Della calms down from crying over the small amount of money she has been able to save. Later, it is suggested that "Perhaps you have seen a pier-glass in an $8 flat" - talking to the reader and encouraging him/her to remember a mirror in such an inexpensive setting looks like.
The conclusion is the most notable example of directly addressing the reader. The author is clearly talking personally to the reader as the moral of the story is presented and explained. "The magi, as you know, were wise men...And here I have lamely related to you the uneventful chronicle of two foolish children"