explain why the Duke uses euphemisms in the poem My Last Duchess.

The duke uses euphemism to explain the way that he had the duchess killed – "I gave commands; Then all smiles stopped together" – shows a facility for avoiding the truth through choice of language. What this could suggest is that the duchess was in fact guilty of greater transgression than he claims, that instead of flirtation, she might have physically or sexually betrayed him. There's certainly no explicit evidence of this, but at the same time, it's plausible that a man as arrogant as the duke, especially one so equipped with the power of euphemism, would avoid spelling out his disgrace to a lowly envoy and instead would speak around the issue.