"My Last Duchess" by Robert Browning is written in the form of a dramatic monologue. As in many of Browning's poems in this genre, the Duke reveals more about himself than he intends as he speaks.
The Duke is often described by literary critics as a "collector", a person who is possessive and whose self-worth is predicated upon the objects he owns or collects. One can imagine that the tour on which he is taking the envoy involves showing off his art collection, something he regards as making himself an attractive suitor by illustrating both his wealth and his good taste.
In showing off the painting of the Duchess, the Duke is doing several things. First, he is demonstrating that he is rich and influential enough to commission a painting by a famous artist. Next, he is showing how beautiful his previous wife was, something again intended to show off his wealth and taste. Perhaps he also intends to explain why he is in the market for another wife, but he quickly moves from a discussion of his previous duchess to his dowry requirements and then to the next work of art in his collection.