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:

I repeat,

The Count your master's known munificence

Is ample warrant that no just pretence

Of mine for dowry will be disallowed . . .

The problem here is just that it's part of an incredibly long sentence. Fra Pandolf is the painter who actually painted the painting of the Duchess:  

Frà Pandolf's hands
Worked busily a day, and there she stands.

The Duke then goes on to explain that he mentioned the name "Fra Pandolf" deliberately ("by design"), because everyone he shows the painting to asks him who painted it.

The complication is in the grammar and syntax: make sure you understand that the sentence below paraphrases "I said "Fra Pandolf" deliberately, for strangers like you never looked ("read") at the picture (and the depth and passion of its glance) without turning to me (because I'm the only one who draws the curtain) and seemed like they would ask me, if they dared, how it was painted.

...I said
"Frà Pandolf" by design, for never read
Strangers like you that pictured countenance,
The depth and passion of its earnest glance,
But to myselfthey turned (since none puts by
The curtain I have drawn for you, but I)                      
And seemed as they would ask me, if they durst,
How such a glance came there; so, not the first
Are you to turn and ask thus.