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.

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.

Friar Laurence’s letter never reached Romeo because its messenger Friar John was stopped by a plague. 

When Romeo was banished for killing Tybalt, Friar Laurence tried to be in communication with him so he would not be cut off completely from his new bride, Juliet. One of the most serious problems that occurred right off the bat was Juliet’s forced marriage to Paris.  Juliet went to Friar Laurence for help, and his solution was to fake her death.  This was only a good idea, though, if Romeo was in on the plan. 

Friar Laurence wrote a letter to Romeo explaining what was going on.  Imagine his surprise when he found out that it never reached its destination.  Friar John arrived, and told Friar Laurence the bad news. 

FRIAR JOHN

Going to find a bare-foot brother out
One of our order, to associate me,
Here in this city visiting the sick,
And finding him, the searchers of the town,
Suspecting that we both were in a house
Where the infectious pestilence did reign,
Seal'd up the doors, and would not let us forth;
So that my speed to Mantua there was stay'd. (Act 5, Scene 2)

Friar Laurence asked the obvious question: Who took the letter to Romeo then?  The answer, unfortunately, is no one.  Friar Laurence was upset, and told him that the letter was important.  He feared that Romeo would do something rash, because he knew Romeo. 

Romeo did do something rash.  He got himself a vial of poison, went to Juliet’s tomb, and poisoned himself.  Juliet awoke to find the husband she had gone to so much trouble to stay with dead by her side.  Devastated, she committed suicide with his dagger.  Naturally, Friar Laurence blamed himself. It is another example of circumstances really being against these two.  They pushed fate and lost.  You could say they had really bad luck.

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.