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.