In terms of narration, the self-effacing style occurs when the narrator is moved to the background. In such cases, the narrator is usually (but not always) not a character. The truly self-effacing narrator is so seemingly absent from the story that the story seems to be telling itself. It's as if a nobody or anybody is telling the story.
Since oral literature was passed down from memory, and countless orators would memorize these stories, some of these older works have the self-effacing style. Homer's Odyssey is a good example. In The Odyssey the narrator simply presents the stories without intruding with his/her interpretation.
Some of Hemingway's work is also characterized by a self-effacing narrator. In his short story "Hills Like White Elephants," there is no authorial or narrator intrusion. The entire story is mostly dialogue. There is no mention of "she thought this" or "he was feeling that," so the interpretation is left to the reader.
They sat down at the table and the girl looked across at the hills on the dry side of the valley and the man looked at her and at the table.
In Homer's Odyssey, the focus is on the story. The same goes for Hemingway's "Hills Like White Elephants," and illustrates his minimalist style. In some cases, the self-effacing style suggests that the reader/listener is given all he/she needs, as if to say, here is the story and the truth lies within, and that's all that is needed. On the other hand, a self-effacing style can be used to suggest the opposite to the reader, as if to say here are some facts and appearances—the rest of the interpretation is left to you (reader).