The question of "why" an author writes a particular work is one of the most enticing yet one of the most difficult to answer. Usually, the only clues to "why" lie in the theme(s) of the story or in the traits of the characters. Sometimes, author biography and contemporaneous social issues might give further indications of "why" as with Dostoevsky and Dickens, for example. In some cases, genre might also give a clue about "why," such as when the literary piece is a satire, which is written to ridicule and reprimand a person, group or society for breaking society's accepted moral and ethical code.
In the case of "The Open Window," we actually have clues from genre, theme, characterization and author biography, which has been adequately speculated about by literary critics. First, the story is a satire of adults who are mentally unstable and/or gullible and foolish. So one reason he wrote this story is to show society a true picture of itself by ironically satirizing the harm being done, presenting mentally unstable Nuttel as both a cause and the victim.
Another clue is the themes, one of which is appearance versus reality. Saki shows how frightfully easy it is for "self-possessed" Vera to deceive and harm the stranger Nuttel and her familiar aunt and uncle, the Sappletons. Appearance versus reality also attaches to the introduction of the supernatural when the husband and younger brothers enter the yard and approach the open window sending Framton (a name indicating one who worries and is mentally unstable) running out the front door. Framton's flight raises a question about what happened to him afterward. This speculative question might be a clue to "why" Saki wrote the story. It certainly relates to what critics call the "enfant terrible":
According to Janet Overmyer, children in Saki's stories often are "cruel to adults because ... [they must] snatch their revenge whenever the opportunity arises." ("Saki's Enfant Terrible in 'The Open Window'." Gale Cengage on eNotes.com)
The enfant terrible introduces clues from Saki's biography. Saki often drew child characters like Vera who routinely and deliberately made havoc in the adult worlds they inhabited. Some critics suggest Saki was drawing a true picture of a type of perverse child and adolescent that he either was or had experience with. Some critics suggest Saki drew the perverse child mind because he never recovered from his oppressed childhood and was taking revenge against his caretakers.
To put all these clues together, it may be that Saki wrote this story because he wanted to improve the adult world he satirized and to warn against the perverse child mind he employed. Perhaps there is another possible way to put the clues together, but there seems to be a good case for this explanation.
"Their bodies were never recovered. That was the dreadful part of it." Here the child's voice lost its self-possessed note and became falteringly human. "Poor aunt always thinks that they will come back someday, they and the little brown spaniel ...."
to mock on the upper class society in that era.
to show out the foolishness of human nature towards little aspects