When Jem and Scout made their raid on the Radley back porch in Chapter 6, they were suddenly surprised by the appearance of a shadow. It was probably Boo Radley, and the children noisily scattered through the gate and the collard patch. A shotgun blast "shattered" the silence of the neighborhood, and the children managed to escape back to the Finch house--minus Jem's pants. Mr. Radley just assumed that it was a Negro who had been prowling about (and possibly stealing collard greens); he knew it wasn't Boo, since Boo must have been present on the porch when Nathan saw the figures running away in the dark. Nathan announced to the crowd who gathered that he "shot at a Negro in his collard patch," and Miss Stephanie retorted with her bad-taste joke about the accused.
"Scared him pale. Says if anybody sees a white nigger around, that's the one."
Miss Stephanie simply--and crudely--meant that the black man who had been shot at was so scared that he turned white.
Jem, Dill and Scout do go onto the Radley property at night to see if they can catch a glimpse of Boo. Jem thinks he sees Boo because he says, "Ar-r" but changes his mind to say "chickens" (52). After peeking in a window, Jem sees a shadow cross in front of him. Later, the consensus is that the shadow was Boo. If that is the case, then after Jem makes a squeak on the steps, Boo could have known it was the kids and told Nathan that it was an adult in the backyard. Nathan goes out and shoots his gun, probably not having seen anyone, and tells the neighbors he was shooting at a black man to justify why he fired his gun.
Once the kids join the adults out front, Miss Stephanie quotes what she heard Nathan Radley say. Stephanie says that it was Nathan who said, "If anybody sees a white nigger around, that's the one." Stephanie goes on to paraphrase by saying, "Says he's got the other barrel waitin' for the next sound he hears in the patch, an' next time he won't aim high, bet it dog, nigger, or—" (54). So Miss Stephanie references a Negro because that is what she claims Nathan Radley said was the reason for his gun going off.
Nathan Radley, it appears, doesn't really know anything about the intruders in his garden. The intruders were simply Jem, Scout and Dill, and it seems that no-one suspects them at all. Mr Radley seems to be under the impression that it was a black man who broke in to steal from his collard patch, and shot at him. This shows the readiness of most whites in Maycomb to believe the worst of the blacks. Mr Radley evidently didn't get a good look at the intruders but simply assumed it was a black man trying to steal.
Not only this: the occasion is turned into a tasteless joke at the expense of blacks, when Mr Radley, according to Miss Stephanie, declared that the shot 'scared him (the intruder) pale.' Miss Stephanie takes this up with gusto: 'Says if anyone sees a white nigger around, that's the one.' The refers to the supposed black intruder turning white from fear. Miss Stephanie, therefore, does not pass up the chance to indulge in a nasty racist joke. This is in keeping with what we see of her character in other parts of the book; she seems to enjoy making fun of others, for instance when she needles Scout during the missionary meeting in a later chapter.
The use of the wholly derogatory word 'nigger' also emphasizes the repellent racism of such people as Miss Stephanie. Sadly, the use of this kind of pejorative language and racist attitude is typical of most whites in the town.