In four short lines, the poem gives us a vicar at a wedding, caught in a moment of quiet vanity. The speaker is proud of his "pretty hands," and that small boast sets the tone. He isn't thinking about holiness or the couple in front of him; he's thinking about how he looks while performing the ceremony.
The phrase "I play a wise man" is the key. He doesn't say he is wise, only that he acts the part. The ritual of giving "husband to wife" becomes a kind of stage performance, and the "two copies of a book" he carries sound oddly modest, almost like props. The mood is dry and gently mocking. There's humor in the way something solemn is reduced to gesture and costume, but the poem doesn't seem cruel. Judging by the voice, the speaker knows he is putting on a show, and that self-awareness makes him oddly likable, as well as a little absurd.