In the story, and this includes Walter's daydreams, the author uses a third person narrator. Using this point of view, the narrator refers to each character as he, she, it, they, etc. "I" is used in first person narration and "you" is used in second person narration. 

Using third person narration, the author has the position of being outside the story. He (the narrator) has a privileged position in that he can describe events from any perspective. He can also describe things in a particular character's mind. This is clearly what is going on with Walter Mitty. 

The narrator is able to go back and forth in describing events in the external world and Walter's daydreams. It would be interesting to consider this story from Walter's perspective (first person), but that might not work because Walter prefers the life of his own mind. In other words, had the story been told by Walter, he could have eliminated any external events or even the nagging of his wife. So, it is possible that Thurber chose third person narration to make the transitions from external to internal (Walter's mind) more significant and pertinent to the story. We need to see how Walter absentmindedly goes through his life as he is distracted by his own daydreams, and this is effectively shown with the narrator's distance from the world of the story.