The narrator in this story is an anonymous, limited third-person narrator. Third-person narrators stand outside the characters and events they describe. Limited third-person narrators, however, are largely confined to giving insights into only one character and do not provide much information about other characters or about anything else.
We can see how limited third-person narration works in 'The Story of an Hour'. We are given the thoughts and feelings of Louise Mallard and no one else; the other characters are described externally. We are given the bare facts of the situation only as they affect Louise: the news of her husband's death, and then the unexpected arrival of her husband, the shock of which kills her. However, the main focus, between these events, is on her meditations as she sits at her window reflecting on how the news of her husband's death has had a sudden liberating effect. She muses how she did feel something for her husband, but nothing like enough to make her grieve for him now, as she values her self-independence much more.
And yet she had loved him—sometimes. Often she had not. What did it matter! What could love, the unsolved mystery, count for in the face of this possession of self-assertion which she suddenly recognized as the strongest impulse of her being!
"Free! Body and soul free!" she kept whispering.
The effects of limited third-person narration are quite clear in this quote. We are seeing straight into Louise's mind here; the exclamation marks denote the strength of her emotions. The narrative at this point identifies wholly with her consciousness; but while supplying her most intimate thoughts to the reader, the narration still stands outside of the character. We might say, then, that limited third-person narration stands midway between omniscient third-person narration - where the narrator knows and relays a great deal of information about different characters and events - and first-person narration, where a character relates his or her story and feelings directly. Limited third-person narration is often more suitable for shorter pieces, like 'The Story of an Hour'.
The use of this type of narration is particularly effective in this story. If we had not had full access to Louise's thoughts, we would wrongly assume, along with the other characters, that she dies simply of shock and joy at her husband's return - as officially pronounced by the doctor. However, because we learn from her thoughts that she felt trapped in the marriage, we recognize that the opposite is more true; she dies not just of shock at her supposedly dead husband's return, but also dismay at the prospect of being boxed in once more. The tone at this point becomes very spare and ironic, quite in contrast to her indulgent musings a little earlier. The contrast between one character's inner life and external appearances in emphasized by the use of this method of narration, and makes it all the more memorable. The narrative approach in this story is therefore quite sophisticated.