A dramatic monologue is the term for when a single speaker goes on and on… and on. This form of writing is certainly what Browning is noted for, since he uses them all over the place. They usually feature a speaker who is in some way a bit funky—set apart somehow from the usual strand of humanity.
Here, we have a slightly naughty and drunken monk spilling his inner secrets about his failings as a monk and his views on art (and how they differ from the Church). Some other noteworthy Browning dramatic monologues include "My Last Duchess" and "Porphyria's Lover".
Browning also has a penchant for dropping us right down in the middle of things without much of an explanation or context:
I am poor brother Lippo, by your leave! You need not clap your torches to my face. (1-2)
See that? Who is Lippo talking to here? Why are torches being shoved in his face? That, young grasshoppers, is some in medias res right in your face. We have no idea what's going on, but we're able to piece it together as the poem progresses.