Here the Duke is showing the painting of the Duchess to the messenger and is explaining that it has been completed very successfully by 'Fra Pandolf'. The Duke normally looks at the painting alone and in private as it is hidden behind a curtain but he is allowing the messenger to take a look as he is so proud of it: 'The depth and passion of its earnest glance.'
The irony is that the painting looks alive when in fact we learn later in the poem that the Duke has ordered her to be killed; he prefers her as a piece of art that he can control. As a painting he can control who looks at her which was not possible in life. The Duke has tremendous power which is why people do not ask questions of him.