Friar Lawrence agrees to marry Romeo and Juliet because he knows Romeo is sincere and he thinks it might end the feud.
When Romeo comes to the friar asking him for advice, Friar Lawrence assumes he has not been to bed because he has been sinning with Rosaline. Romeo is surprised, because he has forgotten all about his old girlfriend. All he can think about is marrying Juliet.
What a change is here!
Is Rosaline, that thou didst love so dear,
So soon forsaken? Young men's love then lies
Not truly in their hearts, but in their eyes. (Act 2, Scene 3)
Friar Lawrence is confused because Romeo seems to have changed his mind so quickly, but Romeo is intense. He decides to go along with it, because he thinks that if Romeo and Juliet get married it might stop the feud.
But come, young waverer, come go with me.
In one respect I'll thy assistant be;
For this alliance may so happy prove
To turn your households’ rancour to pure love. (Act 2, Scene 3)
Lawrence is being overly optimistic and a bit meddlesome here. He knows that Romeo was in love with another girl just the day before. His motives for agreeing to marry them are not completely pure. He thinks that if they are married he can play a part in ending the feud.
Friar Lawrence’s involvement does not stop here. He also gave Juliet the potion to fake her death, and sent a note to Romeo to tell him. Unfortunately, it did not get there.
Unhappy fortune! By my brotherhood,
The letter was not nice, but full of charge,
Of dear import; and the neglecting it
May do much danger. (Act 5, Scene 2)
When Friar John tells Friar Lawrence he could not deliver the note, Friar Lawrence knows that it could be a disaster.
When they both die, he throws himself on the prince’s mercy, explaining that he feels to blame but was just trying to do a good thing.