This poem's in blank verse, which means the lines are in iambic pentameter and there's no rhyming. Don't get too hung up on the fancy vocab. An iamb is just a two-syllable pairing, made up of an unstressed syllable followed by a stressed one. "Pentameter" just means there are five (penta- means five) such units (called feet—let's hope they're not stinky) in each line. Check it out:
I am poor brother Lippo, by your leave!
You need not clap your torches tomy face. (1-2)
This blank verse form allows Browning to recreate through his poetry the ebb and flow of normal conversation. Since it's unrhymed, it doesn't have what we might consider the stilted artificiality that end rhymes would create. For example:
It's natural a poor monk out of bounds
Should have his apt word to excuse himself:
And hearken how I plot to make amends. (341-43)
The first line spills over into the next, which gives us a solid sentence that is uninterrupted by a heavy end stop or a rhyme. This makes the passage sound closer to natural English speech. Instead of rhymed lines, we have sentences that sound more like prose.
People don't, you know, typically speak in rhyme. If they did, we would consider that pretty unnatural (not to mention annoying). So, the blank verse form allows Browning to make Lippo's speech sound closer to a typical conversation than highfalutin verse.
This form is also usually used in epics—Milton gets down with its bad self in Paradise Lost, for instance—but this poem is nowhere near an epic, which gives it a good dose of irony. Since there's nothing really epic about a drunken monk trying to stumble his way home, getting caught by guardsmen, and then giving a long, rambling monologue on his life and his views on art, that's where the irony comes in.