In Shakespeare's Romeo and Juliet, how is Tybalt responsible for Romeo and Juliet's deaths?

Tybalt is responsible for Romeo and Juliet’s deaths because his insistence on fighting Romeo is what got Romeo banished and led to his suicide.

Tybalt was so caught up in his family feud that he did not think about the consequences. He tried to fight Romeo originally at the party, and then caught up with him again later on the street.  Romeo was not interested in fighting. He also considered Tybalt family.

TYBALT

Romeo, the hate I bear thee can afford
No better term than this,--thou art a villain.

ROMEO

Tybalt, the reason that I have to love thee
Doth much excuse the appertaining rage
To such a greeting: villain am I none;
Therefore farewell; I see thou know'st me not. (Act 3, Scene 1)

Mercutio tried to protect Romeo, and Tybalt ended up fighting him.  He could have just forgotten the whole thing when it was clear that Romeo would not fight.  Mercutio was at the party too.  The idea was to eliminate the Montagues or punish the Montagues.  Tybalt paid with his life for his insistence on maintaining the feud.

Unfortunately, after Tybalt killed Mercutio, Romeo had to step in.  He killed Tybalt, and found himself banished.  It was this banishment that led Juliet to fake her death, and this fake death is that which caused Romeo to commit suicide.  He returned from banishment to find Juliet supposedly lying dead.  Unable to accept that, he killed himself.  Juliet then awoke and killed herself.  Thus, Tybalt was directly or indirectly responsible for his own death and the deaths of Mercutio, Romeo, and Juliet.