The townspeople notify Miss Emily's relatives when she begins her courtship with Homer Barron because they disapprove of the relationship and believe it is inappropriate.
Miss Emily is from a wealthy, respected family and the townspeople believe that she should maintain her family's reputation by only associating with people of the same social status. Homer Barron, on the other hand, is a laborer from the North and is not considered to be of the same social class as Miss Emily.
By notifying Miss Emily's relatives, the townspeople hope to put an end to the relationship and preserve the reputation of the Grierson family. This action reflects the societal expectations and prejudices of the time period in which the story is set.