Shakespeare is also crucial to understanding the body of Literature, especially works written in English that came after him. To me, a lot of the reason to teach anything in school is to show students what has come before, so they can understand the foundation upon which the society we now live in was was built. That may seem to be only a question of history class. But it covers every subject imaginable, Literature not the least of these.
Shakespeare obviously had an effect on written works after his day, but also, we should teach Shakespeare because of the effect he had upon our language. He invented over 1500 words that we use commonly everyday. And for that reason alone, students should read and understand his work (and that doesn't include the phrases that we consider cliches that come from Shakespeare's plays -- phrases like: laughing stock, sorry sight, as dead as a doornail, in a pickle and too much of a good thing).
For me, however, it is the breathtaking beauty of his poetry and the complex and intriguing development of his characters that make him a must teach in school.