Why does Framton Nuttel visit Mrs. Sappleton?
Mr. Nuttel visits Mrs. Sappleton because of his exhausted nerves. During the late 1800s, a long recuperative stay in the country was considered a great restorative for one's mental health. Nuttel's sister writes him several letters of introduction and it is through one of these letters that he meets Mrs. Sappleton at her quiet estate. Mr. Nuttel hopes that some time at Mrs. Sappleton's house will restore his nerves so that he can return to his regular life.
In the story "The Open Window" by Saki, the reader is not sure about the exact nature of Nuttel's nervous state or how he got to be this way. Through indirect characterization, Saki explains to the reader that Mr. Nuttel is quite nervous, and this helps to create the rest of the story and Mr. Nuttel's eventual scare at the hands of a mischievous young girl.
Framton Nuttel is not visiting Mrs. Sappleton exclusively. That might be considered inappropriate. He is calling upon Mr. and Mrs. Sappleton, but the men are all out hunting. Only Mrs. Sappleton and her niece Vera happen to be at home when Framton calls. There were no telephones in those days, and there was a lot of time spent calling on people who were not at home. Framton must have planned to arrive at right around tea time, but he was evidently a little early. Vera's practical joke could only have come off as successfully as it did if Mr. Sappleton were gone but expected soon. Evidently Mrs. Sappleton was not expecting Framton's arrival. That explains why she had to send Vera down to talk to him for few minutes while she was getting ready to receive him.
In this story, Framton Nuttel has suffered a nervous breakdown or some other sort of mental issue. So he has been sent out to the country to relax in a less hectic setting. His sister is very concerned that he will not do the things that he needs to do to get better and so she makes sure and sends letters of introduction along with him so that he can meet "nice" people. His sister thinks that, otherwise, he will just be alone and his nerves will get worse.
So Nuttle visits Mrs. Sappleton because she is one of the people that his sister has said that he should visit. He should visit her to prevent his mental issue from recurring and/or getting worse.