During this exchange, Mr. Hooper reveals his own secret sinfulness (without actually naming it as that). To explain the veil, he says,
"this veil is a type and a symbol, and I am bound to wear it every, both in light and in darkness, in solitude and before the gaze of multitudes, and as with strangers, so with my familiar friends. No mortal eye will see it withdrawn. This dismal shade must separate me from the world: even you, Elizabeth, can never come behind it!"
The veil, for Mr. Hooper, symbolizes the figurative veil that each of us wears, a veil that separates each of us from our fellows for as long as we're alive. We pretend that we are sinless, never revealing the secret that we are really sinful, to one another and, thus, remain forever isolated, never to be truly known or understood by anyone. We are all, Mr. Hooper (and, likely, Hawthorne) believes, sinful, and we all try to hide it from each other. We even try to hide our secret sin from God, though a futile act, and this is why Mr. Hooper must wear the veil in company or alone. It is only when we die that God casts the veil aside and our sins are finally revealed.
Mr. Hooper claims that he, "'like most other mortals, [has] sorrows dark enough to be typified by a black veil.'" But the rumors, Elizabeth tells him, say that his sorrow is not "'innocent,'" that he "'hides [his] face under the consciousness of secret sin,'" and she begs him to put an end to the scandal these rumors are causing, especially given the nature of his employment. He says, "'if I cover [my face] for secret sin, what mortal might not do the same?'" At this point, both of them sit quietly. Mr. Hooper has essentially admitted to his fiancee that he does have a secret sin and that it is not innocent; he sins knowing and willfully, just as everyone else does.
No one speaks for several minutes, and finally, the light of understanding fills Elizabeth's eyes: "in an instant [...], a new feeling took the place of sorrow: her eyes were fixed insensibly on the black veil, when, like a sudden twilight in the air, its terrors fell around her." He begs her to stay with him, and she asks that he show her his face once more. He will not, and so she leaves him. In that moment, it appears that Elizabeth has understood the symbolism of the black veil, and that -- in recognizing its significance -- she confirms that she, too, knows the truth that Mr. Hooper conveys in wearing it. The idea that she would always be conscious of the secret sinfulness that separates them and that he is aware of her secret sins (even if he doesn't know what they are) is too much for her.
Ultimately, then, what Mr. Hooper reveals in this scene is the meaning of the veil itself, the fact that its meaning applies to himself as much as anyone else, and his knowledge that he may never directly accuse anyone of sinning in the way he knows them all to do. As soon as Elizabeth realizes the meaning of the veil, she must realize that he knows that she has a secret sinful nature, too, and this makes her too vulnerable; she is too scared to know that someone knows her secret, just as all the congregation would be if they understood the symbolism of the veil. Mr. Hooper is smart enough to know this, but it is a tragic knowledge because it will alienate him from humankind forever. Despite his occupation and dedication as a minister, on some level, he can never fully do his job because the nature of humankind is, first, to sin, and second, to hoard our sin away so that no one on earth can see it.