Earlier on, Faulker tells us that
Colonel Sartoris, the mayor . . . remitted her taxes, the dispensation dating from the death of her father on into perpetuity.
The colonel originally told Miss Emily that he was doing this as a favor to her deceased father, whom he said had given the town a loan some time ago. While Miss Emily believed this, several others evidently saw through it, although they said nothing to her. Emily Grierson had hit financial troubles, so to speak, and so essentially getting rid of her taxes was a way for the colonel to help her keep going without losing her pride.
It's written that Miss Emily would never have accepted a charitable donation. I'd bet that this stems from the fact that the Griersons were a proud and reputable family. They might have once been members of the powerful Southern aristocracy, and Miss Emily likely still had plenty of heirlooms and reminders of that time. The family was described later on - somewhat sarcastically - "the high and mighty Griersons". Therefore, Miss Emily would have viewed it would an insult to have to stoop so low as to accept financial help from others.
The colonel likely intended this as a gentlemen's agreement - informal, not legally binding, but bound by a gentlemanly moral code of the time. The story goes on to say that those who followed him had "more modern ideas", which evidently did not include such agreements. Therefore, they attempted to undo the remission, which, of course, led to Miss Emily's refusal to pay her taxes. We see this inter-generational discord arise in another way, when only a young alderman thinks they should be direct to Miss Emily about the smell emanating from her house.