as far as I can remember, were quite nice."

In those Victorian times there were no telephones. Presumably Framton would just have to appear on the doorstep and hand his letter of introduction to a maidservant. Mrs. Sappleton was not quite prepared to receive a visitor, so she sent her fifteen-year-old niece to substitute for her while she fixed her hair and perhaps changed into a different dress. This gives young Vera an opportunity to practice being a hostess, but she takes a mischievous delight in practicing in an entirely unorthodox fashion. 

Vera is bored to death in this household. She sets Framton up for the scare of his life because she knows exactly what her aunt is going to talk about when she appears, and she even knows that one of the returning hunters is going to sing, "I said, Bertie, why do you bound?" These three men, according to the girl, were supposed to have died three years ago when they were sucked into a bog on the moor. Vera introduces a note of spookiness by telling the nervous visitor:

"Do you know, sometimes on still, quiet evenings like this, I almost get a creepy feeling that they will all walk in through that window."

Mrs. Sappleton is such a rattlebrained woman that it is easy for Framton to believe that her expectation of the three hunters at tea time is a sign of the insanity purportedly brought about by their deaths. Framton never does get to meet the men of the family because he is sure they must be walking dead. The fact that they are all carrying guns makes them even more frightening. What convinces Framton that these men returning towards the open window must be ghosts is the faked look of horror on the face of the girl who had been depicted as "very self-possessed" up to now.

Mrs. Sappleton is the first to see the hunters approaching.

"Here they are at last!" she cried. "Just in time for tea, and don't they look as if they were muddy up to the eyes!"

Framton shivered slightly and turned towards the niece with a look intended to convey sympathetic comprehension. The child was staring out through the open window with a dazed horror in her eyes.

Framton's sister's letter of introduction has just the opposite of the intended effect. Instead of meeting a nice country family, Framton believes he has entered a house of horrors and flees for his life. No doubt he will not be presenting any more letters of introduction to strangers.

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.

"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."

     Framton wondered whether Mrs. Sappleton, the lady to whom he was presenting one of the letters of introduction came into the nice division.

You must first understand that four years ealier Framton's sister had gone to visit people in that part of the country. She therefore knew a number of families in the neighborhood. She wrote letters of introduction to those she did know so Framton might visit them when he went there. One may not visit socially without a proper form of introduction from a mutual acquaintance.

Secondly, you must understand that for some reason we are never told Framton has a nervous disorder that is significant enough to be sent to the country to rest and recuperate. Thus, Framton is going to rest in the country in the same neighborhood his sister stayed in.

Finally you must understand that Framton's sister, whose name we are never given, believes that if Framton doesn't get out and visit people he will dwell unhappily all alone all the time and, in so doing, will worsen rather than better his nervous disorder. Now we can say that the reason Framton Nuttel visits Mrs. Sappleton is that his sister told him to and gave him a letter of introduction because she and his sister had been acquainted, and Framton was in the country to recover from his nervous illness--but--is not permitted to stay alone all the time.   

In Saki’s short story “The Open Window,” the neurotic, gullible Framton Nuttel goes to stay in the country, where the Sappletons happen to live, in the hopes of curing his nervous condition. As a high-strung, anxious man, Framton has been ordered to seek the peace and quiet of a “rural retreat” by his doctor, who believes that the calm, natural atmosphere of the English countryside will ease Framton’s mind and relieve his nerves. Since Framton is not familiar with the region and knows no one in the community, however, his sister gives him formal letters of introduction to his new neighbors. She stayed in the area herself four years ago and vaguely recalls several of the people she met there as being “quite nice.” Moreover, she doesn’t want her brother to spend his time in the country “moping” alone and thus worsening his illness.

Unfortunately, when Framton arrives at the Sappleton home to pay Mrs. Sappleton a formal visit, he is first introduced to her “self-possessed” teenage niece, Vera. After Vera asks Framton several questions, she recognizes that he is an easily excitable man who is completely unfamiliar with his surroundings or with her family and would certainly fall for a humorous prank. Vera proceeds to tell Framton an unsettling story regarding her aunt's open French window, knowing that her uncles are on their way back from shooting snipe on the moors. When the men arrive, Framton mistakes them for ghosts and sprints out of the house. Ironically, Framton's supposedly restful visit to the countryside has made him more anxious and frightened than ever.

Davmor's answer below was spot on! To add, the fact is that Mrs. Sappleton simply was not interested in paying attention to him, the way one normally would in a proper, Victorian visit.

Let's briefly look at this point, in context.

Check out chapter 22 of Emily Post's book Etiquette in Society, in Politics, and at Home. This is one (of several) "guidebooks" on Victorian etiquette as it pertains to guests and unknown people coming to someone's country estate requesting a stay.

"The Open Window" was published in 1914, which takes place way past the Victorian period (1837–1901). However, several social norms from the Victorian era still remained alive then. Shockingly, some others still remain alive, to this day, mainly because the super-influential Victorian era falls right in between the Georgian and Edwardian monarchies. Undoubtedly, the social rules of this era would have taken a lot of time to change with the influx of a series of new, historical circumstances to come.

Framton

All this said, let's focus on Framton Nuttel.

Framton is, essentially, a mental health patient who had a nervous breakdown and currently needs a place to rest properly.

Unfortunately, these were the days before mental health facilities, rehab programs, or even Holiday Inns. Think about this: Sigmund Freud's breakthrough writing The Ego and the Id was published in 1923; this is 9 years before Saki's "The Open Window." Imagine that!

So, now we have a mental health patient existing in a historical time period where mental health is not acknowledged the way it should be. He is seeking treatment that does not yet exist. Then, there are those pesky, old-fashioned, Victorian rules of decorum that he still needs to abide by.

Enter Mrs. Sappleton

As a properly-bred lady of her time, Mrs. Sappleton is not truly (she may pretend, but is not) interested in the maladies and issues of people below her social standing (if Nuttel had his own money, and status, he would have removed himself to his own country estate).

The fact is, she does not care. Framton is a total stranger to her. Plus, she has it all! She has her husband, her family, her niece is there tending to guests, and Stappleton also has the means to afford a country estate. Why bother with the nervous man coming to her for help?

She rattled on cheerfully about the shooting and the scarcity of birds, and the prospects for duck in the winter. To Framton it was all purely horrible.

What happens next is a careless, one-sided conversation between Nuttel, who is trying hard to get her attention, and Mrs. Sappleton, who is eagerly awaiting for her husband and brothers to come back from hunting.

He made a desperate but only partially successful effort to turn the talk on to a less ghastly topic.

Here is the evidence from the text that points to your question's statement that Framton: "feels that Mrs. Sappleton was not giving him his whole attention." The answer is: He does not "feel it"; he totally knows it.

He was conscious that his hostess was giving him only a fragment of her attention, and her eyes were constantly straying past him to the open window and the lawn beyond.

Therefore, adding all facts, we get that we have:

• A fortunate lady enjoying life in a country estate, abiding by the Victorian rules of decorum as much as she can.
• An unfortunate man, nervous and broken, seeking help in the only place his sister could thing of, after all, she wrote his presentation letter.
• A niece, Vera, who is mature and cunning, regardless of her age, and totally messes with Framton's mind.

All of these things conspire together to make Framton realize that he is not wanted there and he is not where he should be.

Mrs. Sappleton has just entered the living room, where Framton is being entertained by Vera. She's late for her appointed meeting with Framton and very apologetic. But Vera's aunt doesn't give the impression that she's particularly interested in chatting with Framton. She rattles through some glib pleasantries, treating her guest to a little talk on shooting and the scarcity of birds. Although she's talking to Framton, Mrs. Sappleton isn't giving him her undivided attention. As she talks, Mrs. Sappleton is constantly looking past Framton to the open window and the lawn beyond. Framton tries manfully to regain her attention by regaling her with the minute details of his ailments. But Mrs. Sappleton isn't remotely interested and only just manages to stifle a yawn. Her concentration remains fixed upon the open window and the lawn outside, where something's about to appear that will terrify poor old Framton.

Framton Nuttel is a highly-strung young man. He suffers from a nervous disorder which is apparently so serious that his doctor has advised him to spend some time in the country for some much-needed rest and relaxation. Framton's sister is also worried about his condition. At the same time, she wants her painfully shy brother to come out of his shell a little bit and meet some people. So she gives him a list of introductions to local families in the area where he'll be staying. One of these families is the Sappletons. The Sappletons live in a large country house. They're a respectable local family, just the kind of people that Framton's sister thinks would be perfect for him to meet and get to know. Who knows, paying them a visit might even do wonders for his nervous condition.

Framton Nuttel is apparently a hypochondriac. He has consulted several doctors in London and they have found nothing physically wrong with him. They have suggested that he go to the country for a rest cure or nerve cure. This was about all doctors could prescribe in older times. Nowadays he would undoubtedly be consulting a psychiatrist in the city.

Framton's sister once stayed in the same pastoral region to which Framton is going. She gives him letters of introduction to various people she met while she was staying at the local vicarage about four years earlier. 

"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."

The Sappletons are one of the country families to whom a letter of introduction is addressed. Framton is obviously not keen about intruding on this family with a letter of introduction from a woman they probably only vaguely remember after four years; and Mrs. Sappleton is obviously not keen about having to meet a perfect stranger from the city. Framton's sister must have assumed that a family living in such a dull place as the English countryside would be pleased to meet someone from the big city. But this is not the case. Mrs. Sappleton finds Framton boring, and Vera considers him such a creep that she takes malicious pleasure in playing a diabolical practical joke on him.