Why does Framton Nuttel feel that Mrs. Sappleton is not giving him much attention?
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.
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.