There are numerous points one could raise in detailing Shakespeare's greatness as a playwright. I'll make a few points.
He added words to the lexicon of the English language like no one else has ever done. Words like amazement, bedroom, bump, critic, exposure, lonely, pious, and useless were probably all first written down by Shakespeare. Not to mention phrases that have become so common they're now cliches: "Too much of a good thing," and "Come full circle," come to mind.
He is a product of his time, as all writers are, but he transcended what any of his contemporaries accomplished. Others were writing revenge tragedies in his day, as many writers had done before him. But he reaches a sophistication and complexity that is unmatched. His word play, his use of motifs, his themes, his characterization are all superior to what anyone else was doing at the time. Shakespeare takes a common plot (Hamlet's revenge) and turns it into a complex masterpiece, for instance.
His vocabulary, his phrasing, and his transcendence make Shakespeare a great playwright. I'll leave the rest for other editors.