How did Lord Voldemort die in Harry Potter and the Deathly Hallows?

Harry was actually the master of the wand Voldemort was using, the Elder Wand. So when Harry casts "Expelliarmus"(Disarming Spell) at the same time Voldemort casts "Avada Kadarva"(Killing Curse), Voldemort's curse rebounded off of the Expelliarmus and killed him instead.

"The bang was like a cannon blast, and the golden flames that erupted between them, at the dead center of the circle they had been treading, marking the point the spells collided. Harry saw Voldemort's green jet meet his own spell, saw the Elder Wand fly high, dark against the sunrise, spinning across the enchanted ceiling like the head of Nagini, spinning through the air toward the master it would not kill, who had come to have full possession of it at last. And Harry, with the unerring skill of the Seeker, caught the wand in his free hand as Voldemort fell backward, arms splayed, the slit pupils of the scarlet eyes rolling upward."

Source

Rowling, J.K. Harry Potter and the Deathly Hallows. New York: Scholastic, 2007: 832.