Shakespeare uses the idea of fate in his works. Fortune is another term for fate. Fate has controlled Romeo, and after he has passed this major turning point in the play, there is nothing he can do. He is doomed, and has become "fortune's fool." His internal conflict is multifaceted. Tybalt was his cousin by marriage. He had no desire to fight him. He wanted no harm to come to him. However, Tybalt fatally stabbed Mercutio. Romeo forgot all about being "related by marriage" and rashly took the life of Tybalt. He didn't want to kill him, but he was so angered by Mercutio's death that he lost control.
Now adding to that frustration is the fact that he will be banished, and his marriage to Juliet will not be coming out for all to hear. His future is spiralling out of control, and he is torn for what he should do. He also must wonder how Juliet will react to his slaying Tybalt. All of these emotions are stirring inside his head, which is what makes up his internal conflict. It's just a very complex internal conflict.
Although Romeo may seem mature, in many ways he is not. Consider, for example, how Friar Laurence admonished him earlier for falling in and out of love so quickly. He is still just a boy. So, when he sees his best friend Mercutio and his enemy/new cousin Tybalt at the beginning of Act III, scene i, he wants to be "a man" in the eyes of his friend, but he wants to keep the piece for Juliet's sake. One part of him really does want to fight, and the other part of him doesn't. Hence, he has an internal conflict.
Of course, that conflict is resolved when Mercutio fights Tybalt and is slain, and Romeo in turn kills Tybalt. Therefore, Romeo becomes "fortune's fool" because fate ("fortune") took away everything in his life in a matter of moments. He has now lost his friend, his wife, and most likely, his life. He was enjoying the greatest moment of his life, and that moment -- the marriage to Juliet -- has turned into the greatest tragedy of his life. He really is "fortune's fool."