What is the significance of the Duke's visitor in the poem and what their presence reveals about the events surrounding the poem (context)?

To his guest the duke, says,

The Count your master's known munificence Is ample warrant that no just pretenseOf mine for dowry will be disallowed;Though his fair daughter's self, as I avowedAt starting, is my object.

Therefore, we can surmise that the person visiting the duke is an emissary for a count whose daughter the duke is pursuing. His reference to her dowry makes it clear that the duke is hoping to marry the count's daughter. In fact, he calls her his "object." He means, of course, his goal, but it is a telling word choice: the narrative makes it obvious that he cares a great deal for objects, things that he can own and control, but he cares less about actual people who have wills of their own that might conflict with his. For example, he is quite proud of the portrait of his "last duchess," but he evidently had her killed when he was unable to control her behavior to his satisfaction. He says, "I gave commands; / Then all [of her] smiles stopped together." He refused to address the behavior that bothered him: that she was made happy by small as well as great things, and she did not appreciate him more than she did any other gift given to her. Because he considers that to be "stooping" (and he will not lower himself).
In fact, the person's visit to the duke, presumably to work out the details of the dowry and nuptials, is what prompts the duke to tell the story of his last duchess in the first place. He is clearly showing this emissary around his home, focusing on the works of art that he possesses, one of which is the portrait of his now deceased wife.