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.