Browning's "My Last Duchess" is a dramatic monologue in which the single speaker talks to a representative of his fiance's father.
Ostensibly, the conversation is preparation for negotiations concerning the amount of the fiance's dowry, but, at least in the Duke's mind, his name is of such great worth that money will be no object for the father, and the Duke will get whatever he asks for. Thus, the negotiations are really about the expected behavior of the fiance once she becomes the Duke's wife.
This is why the portrait of the Duke's last Duchess, his now deceased wife, is featured. The monologue is an implied threat that if the fiance doesn't behave as the Duke insists, she'll end up just like his first wife--executed.
The Duke's first wife, as a portrait, now behaves perfectly, according to the warped Duke. She is passive and just hangs around (literally and figuratively), reflecting back on to the Duke. This is what the Duke wants for a wife. He wants the ultimate trophy wife--a beautiful woman to give him and only him all of his respect and attention. He wants his wife to be a snob just like him.
His former wife was courteous and pleasant to other people (as well as the Duke), and found pleasure in little things, like nature. This was insulting to the Duke.
The poem is framed in works of arts: the portrait in the beginning and the sculpture of Neptune at the end. Thus, the theme of art and reality is revealed. Pride, the Duke's, is also treated.