Why does Vera lie in the open window?

Saki doesn't tell us precisely why Vera tells lies in this spare, very short short story, but he does leave clues. Mr. Nuttel, who has come to stay with Vera's family for some rest, isn't a friend, but has arrived with a letter of introduction. This means he is a friend of a friend, so the family is more or less obligated to be hospitable. He "could only talk about his illnesses" and appears to be a bore. Mrs. Sappleton has apparently left Vera to entertain him as she herself wants to put off facing him.
It's not hard to understand how the 15-year-old Vera would feel resentful at this guest being imposed on her and want to get rid of him. What better way than to make up a malicious story that will send him running away in terror? By telling her lie, she both amuses herself and relieves the family of an unwanted guest.
However, she also lies to her family at the end, indicating that this a form of control and enjoyment for her, a habit that goes beyond getting rid of Mr. Nuttel.
Saki enjoyed creating characters who were not all sweetness and light, but quite capable of lying and cruelty.

Vera's an inveterate teller of tales. We find this out in the very last line of the story:

"Romance at short notice was her speciality."

She doesn't just spin a yarn to poor old Framton Nuttel; she also tells her family a likely story to explain Framton's sudden, terrified departure. Clearly, Vera likes telling stories and the attention it brings her. She proves herself to be very good at acting, whether it's in the role of a demure young lady or as the terrified girl gripped with horror as she sees the three men returning from their day of shooting.
We should never forget that Vera's a fifteen year old girl. As such, she still has a huge sense of fun, if a tad immature, and is always looking for ways to enjoy herself. And putting one over on adults is something she obviously enjoys greatly. The question we need to ask ourselves, however, is if this is something she'll eventually grow out of, or whether it indicates that Vera has rather more sinister character traits.