This descriptive phrase serves two purposes--to describe the actual physical appearance of Miss Emily's house, years after its prime time has passed, and to serve as symbol for Miss Emily's declining importance in the town.
Miss Emily's family came from a long line of plantation owners, who, at one point in American history, held a lot of prestige, wealth and importance in a town. Her family practically ran the town, and were like celebrity elites. The house was a grand, beautiful mansion that stood out as one of the most beautiful on the street. However, as time passed, and plantation families dwindled in success and riches because of the changing social climate of the nation, the house faded and deteriorated. It became run-down and decrepit, and the street that it was on also became run-down and seedy. It wasn't maintained because of dwindling money, so it just decayed, slowly.
Symbolically, Emily's family name suffered decay; her family was no longer the most important one of the town, and no longer held the sway and power that it used to. So, the decay of the house symbolizes her family's decline also. Faulkner, in the line above, personifies the house, saying that it is "stubborn and coquettish," symbolizing how even though Emily's family no longer held their prominence in the town, they still stubbornly behaved as if it did. She still refused to pay taxes, her father still refused to marry her off to someone of "lower class." So, their decline was stubborn, because they refused to admit that now, they were just like everyone else in the town, and were no longer important. They still held themselves above others, even though they were an "eyesore"; they were proud, flaunting their previous status, even though it was empty.
I hope that those thoughts helped; good luck!