There are many reasons for this. In light of this, let me name only three of them.
First, Shakespeare is a great writer and he is able to touch a chord in a person's heart. Moreover, his work, Romeo and Juliet, is one of his best. By these literary qualities alone, people resonnate with the characters.
Second, the work is also timeless in its message. In other words, the theme of love and sacrifice is universal. I suspect all people from all times can appreciate the themes of the work. This also makes the characters of Romeo and Juliet popular.
Third, Romeo's love for Juliet and Juliet's love for Romeo is so pure that it is irresitable not to love. I would go as far as to say that all people want to experience a love this strong (without the tragic element).