The Duke is showing off his art collection to his visitor. Like many art collectors, the Duke is more proud of the value of his collection than of the artistic merit. A large part of the value of any work of art is the identity of the artist who created it. Fra Pandolf is not a real artist but a fictitous creation of Browning, as was "Claus of Innsbruck," named in the last line of the poem. However, from the way the Duke mentions the name of Fra Pandolf it is obvious that the artist is supposed to be famous and his works highly valued. The Duke actually speaks the name Fra Pandolf three times in the monologue. This is a sign of his vulgarity and bad taste. It is almost as bad as telling the visitor how much money he paid to have the work done. (The visitor is there to negotiate the marriage of some Count's daughter to the Duke. It is in extremely bad taste for the Duke to be showing him the portrait of his "last Duchess" and then making it plain that he had had her murdered.)
Browning continuously draws a sharp contrast between the Duke's wealth and exalted position, on the one hand, and his character deficiencies, on the other. He poses as an art connoisseur but shows his bad taste in many ways. His nine-hundred-years-name does not make him any more refined or intelligent or sensitive than one of his servants. Browning probably thought it was a shame that so many great works of art should end up in the possession of money-grubbing vulgarians who were only interested in the investment value of the works.
The Duke is apparently very concerned about the dowry his new bride will bring with her. There is a strong probability that he brought his visitor up to this room to impress him with the value of his art collection so that the visitor will report back to his master that the dowry should be commensurate with her future husband's great wealth and expensive tastes. At the end of the interview the Duke says:
The Count your master's known munificence
Is ample warrant that no just pretence
Of mine for dowry will be disallowed . . .