How can "My Last Duchess" by Robert Browning be viewed as an example of a dramatic monologue? Why has the poet chosen to use this form?

A dramatic monologue typically has a couple of key features.  First, it is in poetry format.  If it were not, then it would be a theatrical monologue.  "My Last Duchess" has that characteristic.  

Second, the poem/speech is one person's speech.  It is not a conversation.  The speaker is also an assumed character.  It is not simply the internal thoughts of the poet.  A character is speaking.  In "My Last Duchess" the speaking character is the Duke.  

I think the reason that Browning chose this format is that the format leaves much of the character interpretation up to the audience.  In a normal third person narration, the narrator can tell audience members that a character is "this" or "that" type of character. The narrator can tell the audience what personality traits somebody has.  But by creating a dramatic monologue, Browning has shown readers exactly what the Duke is like.  We get a much better feel for the character, because the poem is spoken through that character.