Personally, I think the reason why Shakespeare’s plays are still so popular is because he wrote about themes that still resonate as strongly with people today as they did at the time they were written. For example, many people today will have to deal with prejudice and family opposition to a relationship (Romeo and Juliet), gender inequality (Twelfth Night) or jealousy (Othello).
Another possible reason why Shakespeare’s plays are still popular today is that Hollywood has kept Shakespeare alive and well for the modern generations through its many, many film adaptations of Shakespearean plays. Countless people today may have first been introduced to Shakespeare through movies like 10 Things I Hate About You (Taming of the Shrew), She’s The Man (Twelfth Night) or The Lion King (Hamlet).