Browning was indeed a poet closely associated with the dramatic monologue; some more famous examples are 'My Last Duchess', 'Porhyria's Lover' and 'The Laboratory'. The reason he often chose to write in this form is because he was interested in peoples' motivations for their actions and using the first person allows the audience into a character's thoughts and feelings.
The poems above all have the theme of murder and using the dramatic monologue form allows the reader to slowly realise what is actually happening which is very powerful.
This form also makes us question whether we actually believe what the character is telling us. For example the Duke in 'My Last Duchess' claims that the duchess was 'too soon made glad' but do we actually believe him? Or was this just an excuse to get rid of her? We can only imagine the reaction of the courtier that he is telling the story to.
It is interesting for Browning to write as though he were a women or a historical character which is more challenging.