Juliet realized that if Romeo did not kill Tybalt, then Tybalt would have killed her husband Romeo. In Act III Scene ii, Juliet states in line 110, "But wherefore (why), villain, didst thou kill my cousin? That villain cousin would have killed my husband. Back, foolish tears, back to your native spring! Your tributary drops belong to woe, which you, mistaking, offer up to joy. My husband lives, that Tybalt would have slain; and Tybalt's dead, that would have slain my husband." Here, Juliet comes to the realization that if Romeo would not have killed Tybalt, then her cousin Tybalt would have killed her new husband. Her love for her husband is greater than that of her cousin. Juliet, although very young, realizes that the bond in marriage between her and her husband is greater than that between her and her extended family member. Therefore, this scene concludes with Juliet giving her ring to the nurse to give to Romeo, as a sign to Romeo that she still loves and supports him in the wake of these events. Later on in Act III Scene iv, we see that the ring brings Romeo much comfort and hope (lines 180-184), as he was in anguish and uncertain if his new bride would still love him after she found out that he had killed her cousin in order to avenge his best friend Mercutio's death (Act III, scene i, lines 88-135).
Although Juliet was very distraught about Romeo killing Tybalt, Romeo was still her husband and she still chooses to be with Romeo no matter waht the consequences.
For Juliet, romantic love was more important than familial love. Juliet lacks maturity as well, and is led by her hormones rather than her head. Her choice is also one of teenage rebellion. Her father has declared that "I think she will be ruled/In all respects by me" (3.4.13-14). And, for her, Romeo has something of the "bad boy" appeal, the forbidden fruit, as he is the son of her family's long time enemies.
Further appealing to Juliet is Romeo's handsome face and winning romantic words. His willingness to put it all in the line for her, death, banishment, exile are all more attractive than her relationship hotheaded cousin (though this is not to say that she did not love him too; she did, just not as much.)
Even though Romeo slayed Tybalt, Juliet still loves him. At first, feels betrayed by Romeo. Then her love for Romeo takes away the blame she felt against him. At first she sides with Tybalt, but when the Nurse starts cursing Romeo, Juliet says, “Blister’d be thy tongue/ For such a wish! He was not born to shame. / Upon his brow shame is asham’d to sit;/ For ’tis a throne where honour may be crown’d/ Sole monarch of the universal earth,” to defend Romeo. Even though she feels betrayed by Romeo, Juliet realizes that her allegiance is with her husband.
I thought that Romeo and Juliet became close because Juliet thought he loved her and she wanted to be with him.