Who encourages Framton to embark on a journey in "The Open Window"?
The character of Frampton Nuttel in Roald Dahl's short story "The Open Window" is a man suffering from an apparent case of a nervous breakdown for which he makes the trip to the country side in aims of getting what the text calls "the nerve cure".
This visit is orchestrated by Nuttel's sister, who insists that if her brother remains in the current state that he is in, his condition will get worse. Hence, it is she who contacts the country relatives and even sends recommendations for him to take with him as assurance of his "good company" as they would say in those days.
"My sister was staying here, at the rectory, you know, some four years ago, and she gave me letters of introduction to some of the people here"
Therefore, Frampton's sister had already been a guest at the household of the Sappleton's and, for this reason, her word would have been considered as honorable in terms of her suggestion that they take her brother in for a prolonged period of time.
In 'The Open Window' by Saki, where he brings up a frame narrative story, the arrival of Framton nuttel to the Sappleton household can be called the central incident. About your question, it is exactly Framton's sister that has encouraged him to meet the people she knows, for she might have thought loneliness would make his nerves worse. (Since he is suffering from a nervous breakdown.)
"I know how it will be," his sister had said when he was preparing to migrate to this rural retreat; "you will bury yourself down there and not speak to a living soul, and your nerves will be worse than ever from moping. I shall just give you letters of introduction to all the people I know there. Some of them, as far as I can remember, were quite nice."