There’s rather a lot more to be said than has been mentioned yet.
I. Shakespeare has a great poetic gift, it’s true. But he is also the first great master of character drawing. Not until Samuel Richardson opened the door to the modern novel was there anyone else to touch him. Hamlet, Falstaff, Lear, Kate, Beatrice, and hundreds of others are real people in a way that no writer of fiction or drama had ever achieved before, and few have achieved since.
II. He also has another trick. It is hard for a modern to appreciate without study, because Shakespeare’s language is four centuries old, but once one has mastered Early Modern English, his ability to write dialog in brilliant verse that nevertheless sounds as though the characters are making it up as they go along is stunning.
III. It is true that Shakespeare is sometimes unclear because he’s trying to communicate something that goes beyond words. But he often succeeds beyond hope.
…My dear lord!
Thou art one o’ the false ones.
– Cymbeline, III, vi
[T]here’s a special providence in the fall of a sparrow. If it be now, 'tis not to come; if it be not to come, it will be now; if it be not now, yet it will come: the readiness is all:
– Hamlet, V, ii
Biron: Our wooing doth not end like an old play;
Jack hath not Jill: these ladies’ courtesy
Might well have made our sport a comedy.
Ferdinand: Come, sir, it wants a twelvemonth and a day,
And then 'twill end.
Biron: …That’s too long for a play.
– Love’s Labor’s Lost, V, ii
Titus: When will this fearful slumber have an end?
Marcus: Now, farewell, flattery: die, Andronicus;
Thou dost not slumber: see, thy two sons’ heads,
Thy warlike hand, thy mangled daughter here:
Thy other banish’d son, with this dear sight
Struck pale and bloodless; and thy brother, I,
Even like a stony image, cold and numb.
Ah, now no more will I control thy griefs:
Rend off thy silver hair, thy other hand
Gnawing with thy teeth; and be this dismal sight
The closing up of our most wretched eyes;
Now is a time to storm; why art thou still?
Titus: Ha, ha, ha!
Marcus: Why dost thou laugh? it fits not with this hour.
Titus: Why, I have not another tear to shed:
– Titus Andronicus, III, i
IV. The plot of The Tempest is generally believed to be original (though he took a great deal of the technical detail from accounts of a real contemporary shipwreck in Bermuda).
V. I strongly recommend Asimov’s Guide to Shakespeare. Shakespeare was not deliberately trying to be obscure, or (most of the time) high-brow, but he did assume that people knew the kind of stuff that everybody knows (things like who the Simpsons are and why “Free AOL drinks coaster” is funny), and four hundred years have made great changes in that area. Asimov’s book is full of explanations.