How does the sub-plot involving Jessica and Lorenzo support the main plot of Shakespeare's The Merchant of Venice structurally and thematically?
Structurally, Jessica's romance with Lorenzo helps drive Shylock over the edge.
When the play opens, Shylock already feels hard-done-by. After Jessica runs away with Lorenzo, Shylock takes yet another hit, both in his pocketbook and in his self-respect. Jessica takes some money with her, which upsets Shylock. He also, of course, feels betrayed that his daughter left him, and her heritage, to marry a Christian. To make matters worse, everyone is laughing at his frustration. In Act 2, Scene 8 we see this exchange:
As the dog Jew did utter in the streets:
'My daughter! O my ducats [money]! O my daughter!
Fled with a Christian! O my Christian ducats!
Justice! the law! my ducats and my daughter!'
Why, all the boys in Venice follow him,
Crying his stones, his daughter, and his ducats.
Shylock has become a laughingstock. In Act III, Scene 1, we see he blames Salerio, Solanio, and Antonio for this turn of events: "You knew, none so well, none so well as you, of my daughter's flight." A few lines later, Shylock gives his "Revenge" speech. If there was any question before about whether Shylock would really take his pound of flesh from Antonio, his resolve to do so is now hardened.
Thematically, at least three things are occurring.
First, Jessica's subplot confirms Shylock is not a gracious person. He is not even kind to his own daughter. He treats her harshly. She sees nothing attractive in his lifestyle, and is only too eager to leave it.
This leads in to the second theme, that of the contrast between Shylock's bitter way of life and Antonio's gracious one. Unfortunately, in this play that is also painted as the contrast between life as a Jew and life as Christian. Of course, this is an exaggeration and a stereotype at best. The underlying point is that it is better to live a life of giving and receiving mercy than a life of giving and receiving only one's rights under the law. In the play, Jessica is a gracious young woman, so she shows her good character by preferring a merciful life.
Finally, Jessica, like Portia, is a funny, clever, and strong-minded young woman who dresses as a boy and sneaks out of her house to do some good. Her behavior echoes Portia's, but in a minor key. Jessica is not quite as brave as Portia (she is afraid to be seen in the pageboy's costume), nor quite as selfless (she is running away mostly for her own benefit, not to help another). Still, between these two remarkable young ladies, Shakespeare shows a smart and spunky girl can come from any walk of life.