In explaining the meaning of the black veil, start with the meaning of its two parts. Blackness refers to darkness, both literal and symbolic. It refers to night, death, sin, depression, and loss. A veil hides someone and sets him or her off from the world. Combine them, and you get someone who is set off from the world through a barrier of pain, loss, sin, and death. For some in his congregation, this produces, we are told " a feeling of dread." People tell stories about him.
However, Hawthorne notes, "Among all its bad influences, the black veil had the one desirable effect, of making its wearer a very efficient clergyman. By the aid of his mysterious emblem--for there was no other apparent cause--he became a man of awful power over souls that were in agony for sin."
He knows the dark, and so can counsel those there.