Conversational and Mysterious
Anderson uses a simple, conversational style to tell his story. In many ways, it feels a lot like someone just talking to you, complete with side-notes, explanations, and sudden jumps in time.
But the narrator doesn't seem interested in giving everything away immediately—he'd rather slowly dole out information. That's why he starts out ambiguously, first talking about the concept of old women like Mrs. Grimes and then diving into her particular story. That's why he slowly leaks out information about his own connection to the events. Anderson wants to make sure that the reader is sucked in by the mysterious nature of the events, just as the narrator is.
Besides that, however, Anderson writes relatively simply. Not simple in a bad way, but in a "he chose every word very carefully" kind of way. You're not going to find crazily structured sentences or multipage paragraphs in this story—there's not even much dialogue at all. By using this simple style, he's breaking down the barriers between the reader and the story, giving us a closer relationship to the things that are happening. When this is combined with the mysterious approach mentioned above, it leads to a story that feels both grounded and free-floating. That's a unique thing to achieve.