The climax of this wonderful little story is when Walter Mitty faces the firing squad. However, what exactly does that mean? It's not fully clear due to the nature of Thurber's story. To be specific, Mitty's fantasy life is much richer than his dull, limited, and henpecked daily existence. He regularly escapes into fantasy, and that's what he does at the end when his wife takes too long.
In doing so, he essentially commits himself to living in fantasy, rather than reality. Now, there is some ambiguity as to what that will mean here. Sometimes there's a close connection between fantasy and reality (the sound of one engine becomes the sound of another), but it's not fully clear how close it is here. It's possible that the firing squad is all in his head—but it's also possible that this is Mitty's wife "firing" questions and accusations at him.