Going to find a barefoot brother out,(5)
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,(10)
Seal'd up the doors, and would not let us forth,
So that my speed to Mantua there was stay'd.
Friar John had been sent by Friar Lawrence to deliver the letter to Romeo in Mantua. Before Friar John had left the city of Verona, he had gone to find another Franciscan friar to "associate" with, that is, to accompany him on his journey to Mantua. At that time, the "searchers", which were a kind of medical-police watch of the town, suspected that there was plague in the house of the other Franciscan friar. So, as a precaution to prevent widespread infection, they sealed up the doors and prevented anyone from leaving the house. Friar John was thus trapped inside, and unable to leave the city and get to Mantua.
As the reader would surmise, this is event on which the entire plot hinges in this last act. Since Friar Lawrence had someone else deliver the letter, he lost control over the timing of the delivery. The letter contained vital information for Romeo: namely that Juliet was not dead, as he would hear shortly, but had taken a potion which would make her appear dead for a period of time. The knowledge of this would make a huge difference to Romeo; it would prevent him from taking rash actions, such as coming back to Verona (or killing himself.)
This event is made all the more tragic by the fact that previously unknown characters (Friar John and his friar friend) are the ones at fault for the failure to deliver the letter. The decision by Friar Lawrence to have Friar John deliver the letter was a fatal one, although made in all innocence. The fact that the two friars were shut up in a plague-suspected house by city authorities absolves them of any guilt in the matter, for there was no way they could have left and not risked arrest or death. The guiltless nature of these characters in this affair shows to the audience how elaborate deceptions can so easily go awry.