Why are the mayor's last words to Aurelio Escovar important? It's important because he is openly admitting his corruption to the dentist. As my colleague above states, the mayor equates his own power to that of the town's. The mayor represents the corrupt politicians who wield the power of the state as their own.
There may also have been a note of warning in the mayor's words. When Escovar's son tells him that the mayor will shoot him if the dentist does not pull his abscessed tooth out, Escovar is unperturbed. He has his own gun. However, when the mayor comes in and Escovar realizes that the mayor has been suffering for five days, he puts the gun away. The mayor is in no condition to shoot anyone, least of all the dentist who can take away his pain.
It is evident that Escovar knows all about the corrupt ways of the mayor; before he yanks out the infected tooth, he tells the mayor
"Now you’ll pay for our twenty dead men.”
Escovar tells the mayor that he will have to pull his tooth without anesthetic because it is abscessed. Anesthetic is extremely sensitive to pH balances; the pH surrounding the inflamed area may have been too acidic to render any anesthetic effective. Although well-trained dentists can use a number of other methods to numb the area surrounding the infection, we are told in the beginning of the story that our dentist does not have a degree in dentistry. So, why does the mayor go to this dentist? It is tempting to speculate that the mayor's power allows him to tolerate Escovar's illegal dental practice in exchange for Escovar's silence on the mayor's corrupt rule.
In his last words, the mayor may have wanted to remind Escovar that he is still very much the top dog in town. Avenging the death of twenty dead men through an anesthetic-free extraction may very well have been the only satisfaction Escovar will ever get as far as the mayor is concerned. The mayor knows he will never be held responsible for the deaths as long as he is in charge. Absolute power corrupts absolutely.
The mayor's response tells you a lot about the authority he holds. If the dentist sends the bill to the town, the town will pay for it. If the dentist sends the bill to the mayor, apparently the town will also pay for it. The mayor's response shows us that the town will pay all of his expenses, which gives us insight into the level of power he has (a high level).
The response also shows the mayor's attitude at this point in the story. His refusal to look at the dentist when he says it, and his use of the word "damn," show that he is not necessarily pleased to admit the truth to the dentist. So he knows he has absolute authority, and perhaps he admits some guilt in this situation, but he is resigned to it.