In this poem, the duke addresses the servant of a man, a "Count," who has come to discuss the terms of a wedding to take place between the duke and the Count's daughter.
The duke refers to the Count as "your master," as he is the employer of his auditor. He also speaks of the Count's "fair daughter's self" and her dowry, the sum generally paid by a wife's family to the new husband.
In order for the duke to be searching for a new duchess, his last duchess must no longer be living. The duke is, evidently, a man who likes to be in control and who likes to have his pride gratified by others. He shows off the portrait of his "last Duchess" alongside other pieces of art he owns, as though she is just another object in his collection. He always wanted to be able to direct to whom and when she ought to smile and blush, and now he can, because he has hidden her portrait behind a curtain which "none puts by [...] but [him]." In other words, no one else is allowed to draw the curtain and see her face, her beauty, and her blush unless he allows them to. He has finally gained the control over her that he long wanted but would not "stoop" to explain to her. He wanted her to value his gift of an ancient name over all the other much less valuable gifts she received.
Therefore, it makes sense that when the duke mysteriously says that he "gave commands; / Then all smiles stopped together." Readers can assume that he had her killed so that he could start again with a new wife, which is what he's doing now. In the next line, he says, "There she stands / As if alive." Thus, we know that she is dead, and he seems to be responsible for it.