Hawthorne's argument—is that everyone has secret sins within their souls and that by keeping these sins from view, people are not being fully honest with one another. Were everyone to make plain their secret sins and sorrows, they would withdraw from one another, much as the parishioners have done to the minister, who is only being honest.
Within the story, Hawthorne uses a few literary devices to hammer his point home:
Elizabeth describes the veil covering the minister's face as such when trying to convince him to remove it: "Come, good sir, let the sun shine from behind the cloud." The veil is viewed as something miserable and awful like a "cloud" which hides the sunny minister himself. People view the veil as something apart from the minister and not as anything intrinsic to his heart.
When speaking with Elizabeth, the minister alludes to Judgment Day, a biblical future event when all mankind will account for individual sins. This allusion relates greatly to Hawthorne's argument that everyone has secret sins they will one day have to reveal, even if it is in the next life.
Interestingly enough, the figure of the minister himself is much like an oxymoron, where two seeming opposites are united. The minister hides his face, calling attention to secret sorrows and sins, yet he is the most open and honest character in the story. All the other characters hide their sins and do not want to call attention to them, yet they believe the minister is more secretive and disapprove of him all the more, only because he is honest about the existence of the dark part of his soul.