The narrator provides the line. It is part of the narrative; it is not dialogue. "She" refers to Connie, a 15 year-old girl rebelling (quietly) behind her mother's back to the realities of the real world that Connie neither believes or spends time contemplating. She thinks she knows how the world works; she has no clue.
While sneaking off to hang out with the older kids, "on the other side of the tracks," Connie unwittingly draws the attention of a dangerous man, Arnold Friend (the "he" in the piece of narrative mentioned in the question), who appears much younger than he is.
Connie notices him, but sees him only in passing and forgets all about him. He does not forget about her, and shows up one afternoon at her home when no one is around.
The context of the quote is how the two of them converse when she answers his knock to see him on the other side of the screen door.
In terms to what the quote, "She spoke sullenly, careful to show no interest or pleasure, and he spoke in a fast, bright monotone” means to the rest of the story, it describes the manner in which she ends up so totally under his control by the end of the story, with no chance of deliverance. Her conversation with him doesn't encourage, but neither does it discourage him. In fact, it allows him to get closer, invading her psyche. She may be hesitant, but he is the aggressor. He chooses words and a tone that are energetic and non-threatening, but the monotonous tone he uses mesmerizes her into a dark, frightening, hopeless place. By the time he begins to threaten her and her family, she is completely out of her league, and she shuts down emotionally and mentally, resigned to her inevitable death at his hands. She moves toward him and through the door like a zombie.
The story is loosely based on the case of an actual serial killer in the Tuscon, Arizona, who lured and murdered several young girls in the 1960s.