If you visualize the portrait as hanging in a gallery, you imagine that the Duchess is residing in the midst of a number of other art treasures. In the picture below, for instance, you will see that the walls are covered with several paintings. The poem clues us in that there is sculpture there as well, as in the poem's last lines the Duke directs his visitor's attention to a fanciful bronze statue of Neptune taming a seahorse. So this is a relatively large and eclectic gallery.
We can probably assume that the quality of the art is very fine, since the Duke's name-dropping of well-known artists (Frà Pandolf, Claus of Innsbruck) shows that he is something of a connoisseur of the art world and is willing to hire the best artists to carry out his commissions. The overall effect of a room full of such masterpieces is bound to be overwhelming, impressing the Count's representative with the Duke's magnificence and power. If the Duke's goal is to ensure that the envoy will be struck with awe and thus less likely to drive a hard bargain over the marriage settlements, he has chosen the right room for it--this display of his shrewdness and worldly wisdom is bound to make an impact. He doesn't have to press the point that he is not a man to be toyed with; the room will do it for him.
The fact that the Duke doesn't like to have to come right out and say things is evident, particularly in his treatment of the Duchess when he claims that he refused to correct her inappropriate behavior because "Even had you skill/in speech--(which I have not)--to make your will/Quite clear to such a one....'E'en then would be some stooping; and I choose/Never to stoop" (35-37, 42-43). Clearly he does have enormous skill in speech, as the perfectly measured iambic pentameter couplets of the poem illustrate (compare, for instance, the rough meter and labored rhymes of the "Soliloquy of the Spanish Cloister"). Did you even notice that "My Last Duchess" is written in rhymed iambic pentameter couplets? Many people don't because it flows so effortlessly, but in fact its structure is absolutely rigid. Browning uses enjambment, or the continuation of a phrase or sentence past the end of a line of verse, to disguise the line endings and the rhyming words, just as the Duke disguises his power and menace under a veneer of hospitality.
So even as the Duke claims a lack of skill, his own style of speech makes his case for him, as he is perfectly aware. This is a man who is so conscious of the effect he has on people that he plans every move he makes, and every word he utters, in advance. He is also a rigid man who plays by very strict rules--rules that he refuses to clearly spell out.
If you'll look at the first note in your Longman, you'll see that poor Lucrezia was only fourteen when he married her, and seventeen when she died. Can you imagine any teenaged girl you know trying to deal with a man like this?
I have placed the Duchess's portrait at the end of the gallery; perhaps the Duke has led the envoy up to it in a casual manner, pointing out other items of interest along the way. The Duchess becomes a mere possession among other similar possessions, an attractive accessory or what today we might call a "trophy wife." Unfortunately, the real Duchess refused to behave as obediently as her husband's collection of inanimate objects, so by retaining only a picture of her rather than the real thing he has reduced her to the subservient object that he requires.
There is still, of course, the matter of the curtain in front of the painting, which according to the Duke "none puts by...but I" (9-10). In this case the curtain would serve as an object of curiosity in a room that is otherwise full of highly accessible material. The Duke further emphasizes his ownership by covering up the picture and hiding it from prying eyes, as he could not do with his over-friendly wife.