Why are characters important in this story?

We must remember that Vera is not a real person but a character in a story. The same applies to Framton Nuttel and Mrs. Sappleton. They were all "cast" to serve a purpose. Saki probably had the germ of an idea for a story. Someone tells a visitor a cock-and-bull story about how three male family members got killed by being sucked into a bog while hunting birds. When these three carefully described men return towards the French window, the visitor naturally takes them for ghosts and flees in terror. It has to be established, of course, that the visitor is a newcomer who knows no one in the region and nothing about the Sappleton family.
But who should tell the visitor such a story? It might be a mischievous boy--but a boy would probably be out hunting with the men. And a boy might be less believable. It could be some senile family member or old servant who sneaks into the living room while the visitor is waiting for the lady of the house. But a mischievous young girl seems like the best choice. She has to be young enough to be mischievous but old enough to be believable. Vera seems like the most interesting character in the story. She pulls off her complicated practical joke to perfection, both as a story-teller and actress. She seems harmless. Who would suspect that such a polite and innocent-looking girl would be capable of making up such a story?
Saki also created two characters who would fit into Vera's scheme. The visitor would be a city man suffering from what is now called neurosis. He is a nervous wreck. This explains why he has come to the country and why he knows nothing about the family. Mrs. Sappleton is a rattlebrained, housebound woman whose life revolves around her husband and two brothers. Vera knows exactly what her aunt is going to talk about when she appears, and the girl also knows exactly what the three hunters will do and say when they arrive at the expected time for tea. In fact, Vera's motivation in creating an uproar may be that she is terribly bored with the maddening monotony of life in this English country household.
Without Vera, "The Open Window" would not be nearly as compelling as it is. We, the readers, are completely taken in by this demure young girl, just like Framton Nuttel. We do not realize that she has been telling him a "ghost story" until after the terrified visitor has fled from the house and is last seen running down the country road. Then we are let in on the practical joke when we read the dialogue between Mr. and Mrs. Sappleton.
"Here we are, my dear," said the bearer of the white mackintosh, coming in through the window, "fairly muddy, but most of it's dry. Who was that who bolted out as we came up?"
"A most extraordinary man, a Mr. Nuttel," said Mrs. Sappleton; "could only talk about his illnesses, and dashed off without a word of goodby or apology when you arrived. One would think he had seen a ghost."