Because Shakespeare lived in a time of social ambiguity much like ours, Shakespeare's plays speak better to a modern audience bewildered by change, ambiguity and senseless violence than they did to the Victorians who couldn't understand why Shakespeare would write about dealing with changes, deception, ambiguity and senseless violence. They didn't get why Cordelia dies so pointlessly. Modern audiences do.
Yes, Shakespeare's plays are still relevant today. They are classics in English literature for their evocative language and dramatic structure. The stories are timeless. Studying and appreciating Shakespeare also helps students develop a certain mental discipline in analyzing the language. They talk about univeral themes and issues which are relevant today. As drama, they offer actors some of the greatest roles. Movies based on Shakespeare plays continue to do well at the modern box office. A lot of the words we use today were invented by him. Shakespeare was the first to define many psychological disorders, personality types, etc. that later psychologist (such as Freud) use today.
While it may appear that a lot has been changed, the essence of Shakespeare has not been lost, it has merely been adapted to make it more appropriate for younger audiences of today.