What is psychoanalysis? How does Robert Browning explain psychoanalysis through his poems?

Psychoanalysis involves understanding the motivations behind our behaviors, especially motivations that are unconscious.
Browning, of course, did not have the advantages of knowing about modern psychoanalysis, but in several of his poems he has the narrator unwittingly reveal aspects of his personality and motivation that he himself is blind to.
For example, in "My Last Duchess," the duke, thinking he is justifying having his last wife killed, actually reveals himself to be an overly controlling and jealous man who could not tolerate the innocent kindnesses his late wife bestowed on the rest of the world. We see—though he himself is not conscious of it—that his wife was innocent and that he himself is a highly disturbed individual.
Likewise, in "Fra Lippo Lippi," the narrator attacks Brother Lippo for his innocent gestures, such as tending to plants, revealing that he, the narrator, is in fact the sick, twisted person.