How many of Shakespeare's plays have you seen?  If the answer is none, that answers your question.  If the answer is that you have seen plays, try reading a summary first so you know the story before you watch it.  Remember you don't need to understand every detail, just the broad strokes.

I think the language of Shakespeare makes him somewhat difficult to access for most of today's youth, but for people who have spent time studying his works, he can be hilarious, witty, inspirational, and dramatic like no one else can be.  His works will make you laugh, cry, ache, and (most of all) think.

To be completely frank, however, there are a certain number of people who probably claim to be "into Shakespeare," as you say, because they think it looks good to other people or because they think it makes them seem smarter.

If you don't understand Shakespeare when you read his plays, you can try to watch some of the film versions first.  I did this when I was in high school, and I have loved those films and plays ever since.  My favorite film version of a Shakespearean play is Kenneth Branaugh's "Much Ado About Nothing."  After watching that movie - and loving it - I found it much easier to understand the play when I read it.

No matter your mood, Shakespeare has a play for it.  If you think today's teenagers have drama in their lives, you should check out some of Shakespeare's characters sometime.

I hope you will continue to explore this idea, and read enough of Shakespeare's works that you will come to understand for yourself why people are so into him.  :)

It's rather difficult to explain.  I think everybody has their own reasons for loving this literary genius.  I know I get almost giddy when i read any of his plays.  He wrote in a timeless fashion.  The ideas he expressed were cutting edge for his time, yet because they focus on such universal issues they still are.  They express issues that we are still dealing with today.  What, for example, teenage girl has not expressed interest in meeting her "Romeo".  Taylor Swift's "Love Story" is based on the ideas in the play.  She retells the story through her song.  There are so many connections between his work and everyday life. 

 Oh, and he did it in all in the most beautiful and powerful of ways.  He is untouchable.