Friar Lawrence is motivated by two factors in Romeo and Juliet. First, he is a monk. Therefore we must assume that he is living for the greater good, for the glory of God, to serve, etc. This is an underlying motivation for him to do anything that would better society. His ultimate goal in helping Romeo and Juliet is to bring the feuding families together. He believes this marriage will do just that.
Additionally, when we first meet Friar Lawrence in Act 2, it is obvious he has had a long time friendship/advisorship with Romeo. He already knows all about Romeo's infatuation with Rosaline, and likely other girls as well. This suggests Romeo has been confiding in Friar Lawrence for some time. When he brings up his new love for Juliet, Friar Lawrence speaks to him like a father-figure and friend. He obviously has seen Romeo bounce from one infatuation to another in the past. His close relationship is confirmed later in the play when Romeo comes to him crying and the Friar says something to the effect of, "Stop acting like a woman." Only a good friend could say something like that.
Likely, Friar Lawrence realizes that Romeo's love for Juliet is different than the other girls, and desires to see his young friend happy and settled down at last. Coupled with his motivation to bring the two families together, Friar Lawrence truly believes he is doing the right thing by helping the young lovers.