Since I was homeschooled for my high school years, my introduction to Shakespeare was somewhat different than the average American's today. I read my first Shakespeare play in a 9th grade Catholic reader. The book was an older text with some brief notes for each Act and some black and white photos from a stage production featuring Katherine Hepburn. The play was A Merchant of Venice. Since I had no previous experience with Shakespeare and was working through it on my own, I found the notes quite helpful and after that play I was hooked. Although in grade school I was quite a good student and an avid reader (having gone through The Chronicles of Narnia, the Lord of the Rings and The Little House on the Prairie), I had never tried anything as classic as Shakespeare (and certainly not read any of the children's versions), but still found the transition rather easy. Much of Shakespeare's "wit and wisdom" are commonly known today because he still is (by the grace of God!) read widely in high schools of America today.
I don't see the need to wait till high school for Shakespeare, particularly with the nice children's versions available. Although I'm not a big proponent of reading children's versions of every classic novel available, William Shakespeare's influence is so good and so widespread, that I believe an earlier introduction can be an excellent thing.
Language - Studying Shakespeare's plays gives us an understanding of the origin and meaning of many words and phrases commonly used today which have their origins in Shakespeare's works.
The challenging language stimulates the mind and the imagination. This challenge can be very satisfying challenge because the stories are creative, enjoyable and sometimes very funny. The stories are meaty and can really make you think. They are rich with discussion material and "learning opportunities". Shakespeare had an amazing understanding of human nature and so his stories can give us a great deal of perspective on life in general.
Catholic perspective - rooted in Catholic philosophy (stories mention Purgatory, etc. e.g. Hamlet) - moral problems and questions wrestled with in the stories....most especially the consequences of sin.
Specific examples from parts of MacBeth...How justifying and committing one sin (in this case a very serious one) usually leads to more and more problems. MacBeth kills the king in order to gain power but finds himself committing more murders and other atrocities in order to cover up his actions. We also see the dangers of "the occult" by how his ambitions were amplified by consulting with the witches. I noticed when reading MacBeth in college (which was probably my third time through this particular play) that MacBeth commits (in order) the crimes punishable in the innermost rings of Hell according to Dante's Inferno (Book One of the Divine Comedy).
Common Experience - Given the wide-spread study of Shakespeare in America's high schools and colleges, his plays form a "common experience" for many people. Examples from Shakespeare can readily illuminate discussions about politics, the faith, various moral topics, etc.
Ideas for Making Shakespeare Accessible:
Shakespeare Readings: Our family has offered Shakespeare nights several times now to a number of local families. We pick a play, assign parts ahead of time and read an entire play over the course of an evening. The parts are assigned a little ahead of time so readers can have a chance to review their parts ahead of time. There is a wide mix of talent within the group, so we try to shuffle the parts around a bit - adults and kids participate together (this has been a nice way to encourage dramatic reading). People show up with a few small props and a little bit of acting goes on (death scenes seem to be a favorite!) but the atmosphere is fairly casual (one reader has to bring her own braille text!). This format, which has run two or three hours with some light refreshments afterwards (little ones - except nursing babies - stay home with babysitters) has turned out to be a really big hit - from grandparents all the way down to the nine and ten year olds!!! It has worked well for us to offer this every 2 or 3 months.
The Bard of Avon from Ye Hedge School
Article: "Shakespeare Scholars Say the Bard was ... Catholic?" from The Catholic Educator's Resource Center
Interesting example of Shakespeare as a tool of "common experience" from a Fox News article
Shakespeare for Children by Charles and Mary Lamb