I don't get this part from "A Rose for Emily": "apron-remitted her taxes."

The part of this sentence set apart by em dashes, "he who fathered the edict that no negro woman should appear on the streets without an apron," is a parenthetical comment that is not integral to the sentence. Take out this part between the dashes and read the sentence without it:

"Alive, Miss Emily had been a tradition, a duty, and a care; a sort of hereditary obligation upon the town, dating from that day in 1894 when Colonel Sartoris, the mayor, remitted her taxes . . ."

The part of the sentence inside the dashes is an aside comment about the character of Colonel Sartoris.

Hope this helps. Faulkner does use some rather complex sentence structures.