Browning's famous dramatic monologue, "My Last Duchess," begins and ends with the Duke showing off a work of art to his silent listener, a representative of his fiance's father. A dramatic monologue features one speaker talking to a silent listener, and in this case it is as if the reader is listening in on a conversation between the two.
Though the poem is written in iambic pentameter and rhymed couplets, it reads as if one is listening to an actual conversation. Browning accomplishes this by using enjambment, the opposite of end-stopped lines, so that attention is drawn away from the rhythm, metre, and rhyme. Making the reader stop at the end of each rhyme draws attention to the iambic pentameter and the couplets, while making the reader read through the end of each line and move naturally to the next line makes the conversation seem natural.
The encasing of the conversation in the showing off of the two works of art--his former wife's portrait and a sculpture of Neptune taming a sea horse--demonstrates the Duke's reason for being unhappy with his former wife. In short, she failed to behave like a work of art.
The Duke values art, not for its beauty, but for how it reflects back on him. He loves the portrait of his former wife. He draws the curtain to show it off, and she looks "as if she were alive." In her present state--dead and just looking beautiful--she cannot smile at a sunset, or be kind to someone who picks cherries for her. She is now--in the present of the poem--the perfect trophy wife, just as he wants her to be. Notice how the statue of Neptune is not primarily pointed out to the listener because of its beauty or value as a work of art, but because it is "thought a rarity," and because it was "cast in bronze for me." These traits reflect back on the Duke, revealing his good taste and influence and wealth and position. That's what the Duke wants from a wife, too.
The Duke's tremendous pride and hubris prevent from even mentioning his complaints to his former wife. To do so, he says, would be stooping, and he chooses "Never to stoop." The idea is that if she doesn't pay him and only him the absolute respect his "nine-hundred-years-old name" deserves, then while correcting her may change her behavior, if she did not naturally treat him as he demands to be treated, then her change in behavior would not be good enough. If she doesn't do it naturally, then telling her to change could not change that.
It's important to realize that the wife did not do anything except respond to courtesy with courtesy; enjoy nature and animals; and show politeness. But the Duke demands that he and he only receives all of her attentions. In short, he wants a trophy wife who behaves like a work of art--stands and looks pretty--and who is a snob.
It's also important to understand that the entire conversation is a veiled threat to his soon-to-be wife--if she doesn't behave as the Duke demands, the same thing will happen to her: he will give "commands," and all smiles will stop together. He will kill her, like he killed his former wife.