Mr. Frampton Nuttel was a nervous man. In the story The Open Window it is stated that he had suffered some form of nervous setback like a breakdown or some sort of disorder, so his sister sent him down to Mrs. Stappleton with a note. Apparently his sister feels that Mrs. Stappleton is a nice person and so she asked him to introduce himself to her with the note that she sent.
Hence, the reason why Mr. Nuttel took the trip to visit Mrs. Stappleton was as a result of his sister asking him to go there to get better after he suffered his breakdown. Later on we find out that Vera had made up a story that made him even more nervous and the poor thing fled away in a haste.