After Romeo kills Tybalt, he states, “I am fortune’s fool!” Explain how this is shown in the play.

In saying that he is "fortune's fool," Romeo refers to his terrible luck. If you think about it, he has had pretty bad luck throughout. First, Rosaline doesn't return his feelings, and she rebuffs him. Next, he falls in love at first sight with a girl he meets at a party, but she turns out to be the daughter of his family's sworn enemy. Then, she falls in love with him and marries him anyway (yay!), but her vengeful cousin, Tybalt, comes after him. Though Romeo tries to refuse to fight, his best friend, Mercutio, gets involved and is killed when Romeo comes between them to break up the fight. Next, his own anger gets the best of him, despite his resolve not to fight Tybalt, and he kills his wife's beloved cousin. I cannot really think of a single instance of Romeo benefiting from good luck, with the possible exception of Juliet's falling in love with him (though this, too, actually leads to more bad luck down the road).
Even after Romeo says this, the bad luck continues. He doesn't get the message from Friar Lawrence that Juliet isn't really dead. Next, he is able to find a corrupt apothecary who is willing to sell him deadly poison. Finally, he drinks the poison just moments before Juliet wakes up from her death-like sleep.