One way to begin to respond to this statement would be to figure out what issues and themes Shakespeare addresses in the play; then, you can assess which of them are still relevant in today's world. For example, Shakespeare addresses the theme of youthful rebellion with Romeo and Juliet's relationship. The lovers know that their dalliance would likely be frowned upon by their feuding families, and rather than compel them to abandon their feelings, the idea that such a relationship would be forbidden seems only to add fuel to their passionate fire. In the garden, Juliet tells Romeo that her family would kill him if they found him there, but he promises that he would rather die by their hands than live a long life without her love. This is one way in which he shows his devotion to her. They defy their families when they choose each other. The idea that teenagers are rebellious is common today as well. In fact, we now know that it is developmentally appropriate for adolescents to break rules, test boundaries, and defy their parents. Most children rebel against their parents on some level, and it is as common now as it was then.
However, a theme such as family honor may not be so well known to us. No one ever says why the Montagues and Capulets feud, but, in the play, the only thing they seem to fight over is family honor: in the first scene, when Tybalt gets angry that Romeo has come to the Capulets' party, etc. The idea of one's family honor being so terribly important is less common to us today: we don't usually see people getting into deadly fights over a slight to their family honor. "Yo momma" jokes are even considered a form of wit!
At any rate, try to brainstorm a list of all the issues and themes that Shakespeare takes on in the play and then work through your list to figure out which ones still seem relevant. Then, you can explore each one (as I've begun to do, above, with rebellion and family honor) and ascertain whether or not it is still of interest to us today. If it is, describe how, and perhaps provide some examples.