The use of an omniscient third-person narrator enables Chopin to tell a complete story that's not limited to the protagonist's point of view. This is key because the opening of the story begins with us readers knowing something Mrs. Mallard doesn't, and because the story ends after Mrs. Mallard has already died. If Mrs. Mallard were telling the story in first person, readers would be exposed to a whole different explanation of her weak heart, and the story would end very differently – and somewhat earlier.
The use of third-person omniscient narrative voice also keeps Mrs. Mallard more sympathetic and understandable. The narrator seems to be excusing her behavior and thought process, or at least providing reasoning for it. For example, look at this description, stated by the narrator, of how Mrs. Mallard cringes away from the approaching feeling of freedom:
There was something coming to her and she was waiting for it, fearfully. What was it? She did not know; it was too subtle and elusive to name. But she felt it, creeping out of the sky, reaching toward her through the sounds, the scents, the color that filled the air. (9)
This makes it seem like it's not Mrs. Mallard's fault she has these feelings – they chase her down. She's helpless to resist them, passive and powerless. Mrs. Mallard is on the verge of thinking something complicated and not very nice – the short version of that would be, she's kind of glad her husband's dead because she gets to be free. Even though freedom's scary at first she's excited about it by the end. If that were related to us in first person, we might think Mrs. Mallard to be selfish or believe that she didn't love her husband. As told by the narrator, though, it seems like Mrs. Mallard is helpless under the greater weight of human truths.