In the second and third paragraphs of the story, the narrator rather indirectly tells the reader about Framton Nuttel's purpose in visiting Mrs. Sappleton. The narrator says,
Privately, [Framton] doubted more than ever whether these formal visits on a succession of total strangers would do much towards helping the nerve cure which he was supposed to be undergoing.
In other words, then, Framton is undergoing a course of treatment, it seems, for anxiety (i.e., his "nerves"). As a part of this treatment, he has been sent away on some "rural retreat." His well-meaning but absent sister has supplied him with letters of introduction she's written to all of her acquaintances in the neighborhood so that Framton can avoid becoming totally isolated while he's away. He is, therefore, paying calls on people whom he himself does not actually know, so that his nerves do not grow "worse than ever from moping," according to that same sister. However, he feels that these visits are actually making him more nervous!