The reasons for encouraging Romeo and Juliet's relationship would depend on who you ask!
Friar Laurence, who greatly encourages Romeo and Juliet's relationship by marrying them, believes in the possibility that if these two are married, their families may cease their feud. When Romeo is banished from Verona, Friar Laurence advises him to go to Mantua and live there in safety- see Act III, Scene III. The Friar hopes that as long as Romeo runs away to Mantua, after some time, he may return safely to Verona and his marriage to Juliet may be made public. Friar Laurence thinks that once the Capulets have moved on from grieving Tybalt, they may be more open to the marriage and be willing to make amends with the Montagues.
Though Friar Laurence has a more "big-picture" reason for encouraging Romeo and Juliet's relationship, and believes it is for the good of the community, others have reasons more close to home. Nurse, for example, simply wants Juliet to be happy. Nurse has raised Juliet from infancy, sharing her bed, playing with her, feeding her, and is most likely the person who knows Juliet best. It is her dream to see Juliet in a happy marriage and, knowing how she feels about Paris, she is eager to help Juliet in her secret relationship. In Act II, Scene IV, she even goes to Romeo to arrange that Juliet will pretend to go to confession later that day so that they may marry.
If one were to ask Romeo and Juliet themselves, I think their reasons for pursuing a relationship are quite plain- they believe themselves to be in love and want to fulfill all that love is supposed to be. Juliet is facing the possibility of an arranged marriage to a man she does not like, and Romeo is fresh from heartbreak, so the two are probably putting some pressure on themselves to "succeed" at being in a relationship.