As the other commenters have mentioned, factually, Friar John's quarantine prevents him from reaching Mantua to deliver Friar Lawrence's letter to Romeo. On a symbolic level, Friar Lawrence's letter is unable to be delivered because fate works to keep Romeo and Juliet apart. In the Prologue, the Chorus calls the couple "star-crossed lovers" who have come into the world from their parents' "fatal loins": this means Romeo and Juliet are fated to be crossed in love, to be incredibly unlucky (or "misadventured"). They have been cursed with unpredictable and terribly bad luck since the night they met; Romeo goes to the Capulets' party uninvited, Tybalt challenges Romeo hours after he secretly wed Juliet, Mercutio dies when Romeo came between him and Tybalt, Romeo kills Tybalt, Juliet's father rescinds his earlier position on her marriage to the County Paris and tries to force her to marry him in two days' time, and now the Friar's letter goes undelivered because of some random illness that has never been an issue in the play until now? This final unlucky situation is just another symbol of the ways fate conspires against this young couple.
As others have said, the letter doesn't get to Romeo because the messenger Friar John was not able to send the letter to Romeo (who is in Mantua) or enter Mantua himself due to the precautions taken by the town to prevent illness, i.e. bubonic plague (also referred to as "the Black Death"). Upon finding out that the letter cannot reach Romeo, Friar Laurence plans to reach Juliet before Romeo arrives. However, this plan fails as well; Romeo gets there first and upon discovering (thinking) that Juliet has died, kills himself.
An interesting side note is that Shakespeare wrote this play around the time that the plague was taking over Europe. In fact, due to extremely high death tolls, the theaters had closed down until the infection lessened. Upon reopening, Romeo and Juliet was one of the earliest plays to be performed at theaters. So the existence of the plague in the play must have resonated very closely with the audience that had just been through it themselves.
In Act V, Scene 2, Friar John is not able to get to Romeo because there was suspicion of a plague and anyone who was trying to enter Mantua was quarantined. No one would take Friar Laurence's letter from Friar John because they were afraid they might catch the plague. Therefore, Romeo never received word that Juliet was in a deep sleep, induced by herbs, and would wake up.
The note from Friar Laurence never reaches Romeo because Friar John, who was to deliver the message, was quarantined in a house because they thought he had been exposed to the plague.
By the time that Friar John is able to relay this information to Friar Laurence, Romeo has already been told by Balthasar that Juliet has died and Romeo has returned to Verona.
Friar Lawerence gave the message to Friar John, but Friar John gets caught in a delay with quarantine.