In Romeo and Juliet, why is Romeo banished?

Short Answer: Romeo is banished from Verona because he participates in an altercation involving his family's feud with the Capulets and ends up killing Juliet's cousin, Tybalt.
In act 3, scene 1, Tybalt approaches Romeo and calls him a villain before challenging him to a duel. When Romeo refuses to fight Tybalt, Mercutio defends Romeo by calling Tybalt the "King of Cats" and drawing his sword. While Mercutio and Tybalt are fighting, Romeo intervenes by getting in between the two men, which allows Tybalt to fatally stab Mercutio underneath Romeo's arm. After discovering that Tybalt has killed Mercutio, Romeo seeks revenge and challenges Tybalt to a duel. Romeo ends up killing Tybalt and flees before Prince Escalus arrives at the scene. When Prince Escalus learns about Mercutio and Tybalt's deaths, he banishes Romeo from Verona by saying,

And for that offence Immediately we do exile him [Romeo] hence (Shakespeare, 3.1.148-149).

Instead of being punished by death, Romeo is exiled from the city he grew up in and is forced to flee to Mantua—before he secretly returns to Verona and discovers that Juliet is presumably dead.

Romeo is banished in Shakespeare's Romeo and Juliet after killing Tybalt in a street duel. In spite of the fact that Tybalt started this duel and killed Mercutio, one of Romeo's dearest friends, in the process, Romeo is the "last man standing" and must bear the responsibility and consequences for the bloodshed. 
At the beginning of the Play, Prince Escalus finds the Capulets and Montagues brawling in the streets and, thus, declares that there will be severe punishment if they are caught fighting again:

By thee, old Capulet, and Montague.
Have thrice disturb'd the quiet of our streets,
And made Verona's ancient citizens
Cast by their grave beseeming ornaments,
To wield old partisans, in hands as old,
Canker'd with peace, to part you canker'd hate:
If ever you disturb our streets again,
Your lives shall pay the forfeit of the peace.

Thus, when the Prince discovers that violence has once again broken out between the Capulets and Montagues in Act Three, the punishment he hands down to Romeo is actually relatively gentle. Rather than make Romeo pay with his life (as he had previously threatened), the Prince merely exiles Romeo from Verona. This is especially compassionate given the fact that the Prince has lost a relative in the duel, as Mercutio was related to him. 
Alas, for impulsive, stubborn Romeo, this punishment--which will result in him being separated from his new wife--is just as unbearable as death.