In the poem, The Last Duchess the story goes that the Duke de Ferrara was planning to take the hand in marriage of whom has been historically linked to be Barbara, sister of the Count of Tyrol. The latter was the man whom the Duke was speaking to, as he was the person in charge of making the wedding arrangements.
Browning waits until the end to expose of whom and to whom he is speaking, because special language had to be employed to describe the beauty and joy that the last Duchess caused on the Duke. With this description, both the reader and his listener would assume the intense love that the Duke felt for his last Duchess, and also helps to accentuate the mourning and loss that the Duke must have felt.
Yet, as the conversation between the Duke and the Count continued, we see a radical change in the speech of the Duke, whom explains how the last Duchess was happy with more than one man, and how he "made the command."
This last comment is what shocks the reader towards the end as we realize that it was the Duke who commanded the death of the Duchess out of anger.
When he immediately switches the conversation back to the Duke and his wedding arrangements to Barbara is when we see that the Count is about to place his sister in a very shady situation, and that gives continuity to the poem. This is a stylistic license that Browning used to wait until the end to bring surprise.