Why is Frampton there and what does it tell you about his character?

Framton Nuttel is calling on the Sappletons because he feels more or less compelled to do so by his sister. 

"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."

She may be the type of person who would have no qualms about calling on strangers herself. But she characterizes her brother as a different type. He is shy, retiring, diffident, reclusive, introverted. He doesn't want to be calling on the Sappletons, and he doesn't want to use all the other letters of introduction his sister foisted on him. His original notion about spending time in the country may have involved living quietly in a little cottage and taking long walks by himself. According to the narrative:

Privately he 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.

We can sympathize with Framton and identify with him, especially since he is the viewpoint character. Most of us would probably feel uneasy about calling upon a succession of strangers and feeling obliged to try to make a favorable impression. The Sappletons would most likely have only the vaguest memory of Framton's sister, who hasn't been back to this part of the country in four years. The Sappleton men aren't present. They may or may not have made conversation easier, as they might have begun to talk about shooting birds. Chances are that Framton, a city dweller, knows nothing about that bloody sport. Eventually it would become all too apparent that he had nothing to contribute.
Like all the characters in "The Open Window," Framton was created to suit the purposes of the plot. He is a complete stranger to this part of England. So he believes everything Vera tells him. He is a bundle of nerves, so he is likely to display a strong reaction to the appearance of the three "ghosts" approaching the open window.