Why do Romeo and Juliet kill themselves?

Romeo and Juliet are fated to die; that much is known from the prologue. And we're also told that they will take their own lives but not why. In act 4, scene 1 Juliet is in dread at her impending marriage to Paris; Romeo is her true love, the man that she wants. She expresses her great sadness to Friar Lawrence:

O, shut the door! And when thou hast done so,
Come weep with me, past hope, past cure, past help.


The friar hits upon a fateful plan—Juliet will drink a sleeping potion the night before her wedding. Then, once everyone thinks she is dead, she'll be laid to rest in the Capulet tomb. This will allow Friar Lawrence to get word to Romeo, and Romeo will return to Verona and take Juliet away with him back to Mantua, where he is living in exile.
It is a potentially dangerous plan, but Juliet still goes along with it enthusiastically. As we can tell from the prologue, however, all does not go according to plan. Juliet drinks the potion and, as expected, falls into a deep sleep, so deep in fact that her family thinks she's dead. Friar Lawrence duly sends a letter to Romeo inviting him to come and fetch Juliet. Unfortunately, his messenger is unable to deliver the letter as he is placed in quarantine due to an outbreak of the plague. Instead, Romeo receives bad tidings from Balthasar, informing him that Juliet is dead.
Romeo, overcome with grief, buys some poison, determined that he will kill himself and lie with Juliet in her tomb. When he arrives to see her, Juliet looks asleep rather than dead—which of course she is, but Romeo doesn't know that—and after bestowing a kiss upon her Romeo takes poison and dies. When Juliet awakes and realizes the full horror of what has taken place, she resolves to join Romeo in death. She kisses his poison-tainted lips in the hope that she too may die the same way but to no avail. Instead she unleashes Romeo's dagger and stabs herself.
Romeo and Juliet are so passionately in love that they literally can't live without each other. They are both equally prepared to die for love as live for it. Their deaths destroy their earthly love, a love forever fated to die. At the same time, their tragic end represents the consummation of a higher love, one that will live on for eternity.