Inscrutable means hard to understand or interpret. For the reader, Walter is not inscrutable. We know why he retreats into his imagination. He is not the stereotypical masculine man, not one who gets things done; and this irks his wife to no end. She is overbearing and a major factor in Walter's continuing practice of seeking solace in his own imagination, but she is also stuck with a man (Walter) who continues to lose focus in the real world in order to focus on his own daydreams.
So, the reader does have a pretty good idea why Walter acts the way he does. However, it does take a bit more thought into why Walter has become this way. One argument that has been made is that Walter lives in a modern time and his career and lifestyle have made it conducive for him to become bored, unchallenged; therefore, he resorts to daydreaming in order to compensate for his boring life. This is a common theme with modern man (and women in other texts) whose instincts command action (hunting, etc.) but living in a modern, civilized world, modern man is restricted to mundane desk jobs, grocery shopping, and so on. So, all in all, Mitty is not inscrutable to the reader.
However, why would Thurber end the story with:
Then, with that faint, fleeting smile playing about his lips, he faced the firing squad; erect and motionless, proud and disdainful, Walter Mitty, the Undefeated, inscrutable to the last.
In this daydream, Walter is inscrutable to the firing squad. They can not fathom why he is so confident and proud. Either he is unafraid of death or he knows that he will escape from this predicament as well, unbeknownst to his captors. Also, in the real world, Walter is inscrutable to his wife. She is always confounded about Walter's absent-minded behavior. Even others around Walter (the parking lot attendant, the woman who overhears him saying "puppy biscuit") don't understand what Walter is thinking/doing/saying.
So, to the reader, Walter is not inscrutable. But to the characters in the story and the characters of his own imagination (who do not know of his incredible abilities), Walter is inscrutable. We get him; they do not.
I would say that Walter Mitty is not inscrutable to the reader because the story depends upon the exposure of his "secret life." However, to the people in his world, including his own wife, he must be inscrutable because they obviously have no idea of what is going on inside his mind while he is ostensibly doing ordinary things like driving a car or buying a box of puppy biscuits. Mitty wants to keep his secret life secret. He doesn't want to share it with anybody. He seems proud of the fact that he is secretly leading many heroic lives. He is like a spy in enemy territory.
His wife keeps showing concern about his physical and mental condition. She may feel more concerned about her own welfare than about her husband's. This seems indicated when she says:
"You know I don't like to go more than forty."
Mitty, despite his absent-mindedness, is probably the breadwinner who makes her comfortable suburban life, with weekly visits to the hairdresser, possible. If she really knew what was going on in his mind she would be even more concerned. She might want him to start making regular visits to a psychiatrist--or she might begin to think privately about having him committed, like the wife in Thurber's story "The Unicorn in the Garden." There wouldn't be much point in Mitty's seeing a psychiatrist because he would probably remain just as inscrutable with the doctor. Mitty's secret life is private and special; it is all that keeps him going.
How many of the millions of men driving cars on the highways and freeways are fantasizing about being racing drivers or fighter pilots? Perhaps the people who design the automobiles even include superfluous interior gadgetry that will encourage fantasies such as this:
The Commander stared at the ice forming on the pilot window. He walked over and twisted a row of complicated dials. "Switch on No. 8 auxiliary!" he shouted...."Full strength in No. 3 turret!"
Mitty is really looking at the various gauges on his dashboard, some of which, such as the oil-pressure gauge, he may not even know how to interpret. He does know how to read the speedometer, and he is probably deliberately accelerating up to 55 mph when his wife cries,
"Not so fast! You're driving too fast! What are you driving so fast for?"
"The Secret Life of Walter Mitty" is James Thurber's best known and most popular story. It has twice been made into major motion pictures. The reason for its wide appeal is probably because many of us can relate to Walter Mitty. Many of us have private lives in which we are able to do things we never do in our public lives. Thurber must have been offering a glimpse of his own secret life in this story, although he may never have had the same fantasies he attributes to Walter Mitty.