Well said by lavincen. Shakespeare took a story premise of young forbidden love and rival families that had been around for hundred of years even by his time. He knew it was a good plot, but he made it more accessible and marketable to the growing audience of his time and future generations.
Mainly, he owned the story through his artistry with the language. Since the plays of his time were not high budget affairs with much else beyond custome and a few props, he had to keep audiences captivated with his language, and his words have stood the test of time above all other contributions he has made. Shakespearean language has worked its way into our own through he witty puns, insightful quotes, and even newly created words and phrases. Now, to be a part of our modern culture, you have to know the language of Shakespeare, which is really pretty incredible when thinking of his influence.
Romeo and Juliet has stood the test of time, relatively speaking, because it deals with the aspect of romantic love and young love, in a modern way that still applies for young people today. Also, since the audience knows how the play will end, the audience can pay attention to how the tragedy unfolds. It is accessible to younger people, because it is about them and their need to be loved is shown along with terrible decision making.
Another reason is this play is read mostly in high schools, it is introduced to a young audience, and as they age, they can look to the play as their introduction to Shakespeare and how they look to the play changes as they age.