Why is Tybalt responsible for Romeo and Juliet's deaths?

Tybalt has a clear and potent hatred of everything that the Montagues stand for. His loathing is most obviously displayed during the Capulets' ball. When he notices Romeo, who has gatecrashed the occasion, he immediately wants to confront and remove him. When he informs his uncle, Lord Capulet, of his plan, he is harshly admonished and feels both insulted and humiliated, first by Romeo daring to enter their private celebration and, second, because his uncle scolded him for wanting to do what he believes is right: honoring and defending the Capulets against a despised enemy.
Tybalt is so angered by Romeo's audacious act that he later sends a written challenge to the Montague household in which he wants the family to answer for what he believes was an insult to the Capulet name. Mercutio tells Benvolio that Romeo will answer the challenge but doubts whether he is man enough to stand up to Tybalt.
In Act lll, scene l, after a verbal altercation between Tybalt and Mercutio, Romeo arrives on the scene and is immediately challenged by the former. Romeo refuses to engage him in a duel and says:

I do protest, I never injured thee,But love thee better than thou canst devise,Till thou shalt know the reason of my love:And so, good Capulet--which name I tenderAs dearly as my own--be satisfied.  

His response is undoubtedly informed by his affection for Juliet, Tybalt's cousin. The hotheaded Mercutio is upset by what he sees as Romeo's weakness and takes up Tybalt's challenge. Romeo tries to intervene when the two start fighting and he inadvertently gives Tybalt an advantage so that he mortally wounds Mercutio. 
Tybalt's exaggerated pride, desire for revenge, and hot-headed petulance are directly responsible for Mercutio's death. After Mercutio's death, Romeo confronts Tybalt and, empowered by revenge, kills him in a duel. This incident has a dramatic effect on the plot and drastically changes the lives of our two protagonists. When the Prince learns that Romeo has killed Tybalt, he banishes him from Verona on pain of death if he should return.
It is Romeo's exile that eventually drives both him and Juliet to desperate measures in order to be together. In an ironic and tragic twist, the convoluted plan, suggested by Friar Lawrence, fails miserably and culminates in their deaths. Tybalt's actions, therefore, set in motion all the events which followed. It is in this manner, then, that he is indirectly responsible for the deaths of the two star-crossed lovers.

In many ways, it could be said that Juliet's hotheaded cousin Tybalt is responsible for Romeo and Juliet's deaths. In Act III, Scene 1, Tybalt challenges Romeo to a duel. Romeo refuses, but Mercutio accepts the challenge in an attempt to protect his friend's honor. Tybalt kills Mercutio, which goads Romeo into combat. Romeo kills Tybalt and is forced to flee the city. Later, we learn Romeo has been banished from Verona and will be killed if he is caught returning. 
Thus, Tybalt is essentially responsible for Romeo's exile, and Romeo's banishment sets up the elaborate death potion hoax that ultimately goes awry and results in the deaths of both Romeo and Juliet. As such, although Tybalt doesn't directly kill Romeo and Juliet, his decision to goad Romeo into a duel is indirectly responsible for the tragedy that unravels in the later acts of the play.