A short story does not necessarily have to teach a lesson. It is a work of art, and as such it should convey what Edgar Allan Poe called an "effect." He meant something like a "feeling," a "mood," or an "emotional effect." The "effect" Saki seems to have been trying to achieve in "The Open Window" is one of amusement or sardonic laughter. However, if we are looking for a "lesson" in the story, it would seem to be based on what is foreshadowed in the second paragraph.
Framton Nuttel endeavoured to say the correct something which should duly flatter the niece of the moment without unduly discounting the aunt that was to come. Privately he doubted more than ever whether these formal visits on a succession of total strangers would do much towards helping the nerve cure which he was supposed to be undergoing.
The poor man is a nervous wreck because of the stresses of life in the big city and has come to the peaceful English countryside for what he calls a "nerve cure." He finds himself in a zany household where a young girl tells him a ghastly story and her aunt appears to be insane. Then three men carrying guns appear in the dusk headed towards the open window, and they are undoubtedly walking dead--ghosts of the men who were sucked into a bog three years ago.
Framton grabbed wildly at his stick and hat; the hall door, the gravel drive, and the front gate were dimly noted stages in his headlong retreat. A cyclist coming along the road had to run into the hedge to avoid imminent collision.
For all we know, Framton could have kept on running until he had run all the way back to London. He has learned the lesson that life is just as stressful in the country as in the big city. Or, to put it another way, people are just as crazy in the English countryside as they are in London. Maybe they are even crazier in the country because they have more room to expand their personalities. The city people have to learn to get along with each other, but there is no such pressure for conformity in the country. Vera really is a little bit crazy. Her aunt is obviously more than a little bit crazy. The three men who can think about nothing but shooting birds and getting all covered with mud are not entirely sane--and they are more dangerous because they are all armed.
Another lesson is that doctors of the period didn't really know very much about neurosis. All they could do was suggest an ocean voyage or a retreat to the peaceful countryside. Framton has consulted many doctors, and all of them have offered the same prescription.
"The doctors agree in ordering me complete rest, an absence of mental excitement, and avoidance of anything in the nature of violent physical exercise," announced Framton, who laboured under the tolerably widespread delusion that total strangers and chance acquaintances are hungry for the least detail of one's ailments and infirmities, their cause and cure.
He comes to the Sappleton household in pursuit of "complete rest, an absence of mental excitement, and avoidance of anything in the nature of violent physical exercise." He gets just the opposite--especially the violent physical exercise!