It would seem that something very subtle is happening in this dramatic monologue. There are a number of people socializing downstairs. The Duke has brought his visitor upstairs ostensibly to show him some of his art collection but really to discuss the matter of a dowry in private. The Duke is planning to marry the daughter of a wealthy Count. Near the end of the monologue he says, "I repeat, / The Count your master's known munificence / Is ample warrant that no just pretence / Of mine for dowry will be disallowed." So he is repeating what he has already told this man, showing that the dowry was uppermost in his mind.
The Duke reveals the painting of his last Duchess to the visitor. We can imagine that he is not looking at his guest while he is talking but looking at the painting, and it is such a good work of art that it brings back memories of the beautiful girl herself. He is really speaking to himself throughout much of the monologue and is practically unconscious of the presence of another man. He says that his deceased wife is "Looking as if she were alive." Because of the hypnotic influence of the painting, with the feeling that he is actually in the presence of his last Duchess, the Duke reveals a lot more about his thoughts and feelings than he intended. Perhaps he even admits to having had the young woman killed without having intended to do so. He is in fact just thinking out loud.
When the visitor abruptly stands up, the Duke comes out of his spell and remembers that they were there to discuss a dowry. But it appears that his reminiscences have had such a negative effect on the visitor that the man will report back extremely unfavorably to his "master" and that the proposed wedding will be called off, as it should be if the Count has any sense and any love for his daughter.
We see that the visitor is hastening to get away. He is starting downstairs by himself without saying a word of thanks or farewell. The Duke has to chase after him saying, "Nay, we'll go / Together down, sir." And when he tries to call attention to the statue of Neptune taming a sea-horse, he seems to be trying to distract the retreating visitor from thinking about the painting of the murdered Duchess and what he has said about her.