The quote is important because it is foreshadowing Romeo’s banishment and the disaster that results.

Foreshadowing is a hint at something that will happen later.  It is an important device because it allows the writer to create a mood and establish a chain of events.

At this point in the play, the prince is angry because the feud between the Montagues and Capulets has gotten out of control.  After a huge brawl in the streets, the prince had enough.  He decided to lay down the law, and give the people an ultimatum. 

Rebellious subjects, enemies to peace,

Profaners of this neighbour-stained steel—

Will they not hear? (Act 1, Scene 1)

Prince Escalus, the leader of Verona, has to maintain order.  When men are fighting in the streets and killing each other over a ridiculous feud, he must do something.  He decides to tell the people that if there is any more fighting, whoever starts the fight will be executed. 

This incident is important because it foreshadows Romeo’s banishment, which is the event that started a chain of events that led to both of their deaths.  Romeo does fight with Tybalt, and kills him.  Because Romeo did not start the fight, he is only banished and not killed.  Unfortunately, that is as bad as death for him because it keeps him from Juliet.

Romeo is not interested in the feud.  Unlike the servants and some other members of the families, he mostly sticks to himself.  He is basically a lover and not a fighter.  When Romeo goes to the Capulet ball, he is not there to start trouble.  He just wants to see his ex-girlfriend Rosalind again.  Instead, he meets Juliet.  Unfortunately, Tybalt sees him there and gets angry.  The feud is important to Tybalt.

Juliet’s cousin is fiery and saucy.  He dislikes Romeo just because he is a Montague.  It is people like him that keep the feud going.  When Tybalt encounters Mercutio in the market, he starts a fight.  Romeo is trying to stop the fight when he accidentally gets Mercutio killed.  He has to fight Tybalt after that, and he is angry and sad that his friend is dead.

Alive in triumph, and Mercutio slain?

Away to heaven respective lenity,

And fire-ey'd fury be my conduct now!(Act 3, Scene 1)

The result of this fight is that Romeo is banished.  Friar Lawrence tells Romeo that he should be grateful that he does not have to forfeit his life as the prince ordered.  For Romeo, he might as well be dead.  Banishment means he can’t be with Juliet, the love of his life.

In this one line, way back in the first scene, Shakespeare establishes a complicated web of tragedy that ends in Romeo and Juliet’s deaths.  The prince makes this proclamation, Romeo sees Juliet at the party, Tybalt sees Romeo at the party, Romeo fights Tybalt, Romeo is banished, Juliet fakes her death, Romeo thinks she’s dead and kills himself, and Juliet kills herself.  All of this was just because the feud caused a fight in the marketplace and the prince declared that anyone who fought would forfeit his life.

Foreshadowing allows the reader (or viewer) to predict what might happen.  It makes us more involved, and helps create suspense.  We know from the prologue that Romeo and Juliet are fated to be lovers, and fated to die.  If we look, we can see how this event foreshadows all of that.