Wow - this could go in so many different directions. First and foremost, he was an excellent writer. Not only was he able to create deep tragedies, light comedies, and meaningful histories, he was able to do so using an extremely strict style of writing. I often challenge my advanced students to see how much of a conversation they can write in iambic pentameter - often, it's not very much! He was a huge wordsmith. Words and phrases that he coined are still in wide use today. There are few other plays that allow for the study of as many literary devices as a Shakespearean play. How many instances of foreshadowing, alliteration, atmosphere, irony, hyperbole, metaphor, etc. etc can be found in any one play? Very few writers are able to successfully write across many genres as WS was. This is why modern film makers are able to still sell redone movies like Romeo and Juliet and even the Taming of the Shrew (10 Things I Hate About You). If all of that wasn't already enough to make him famous, he personally oversaw the building of the most fantastic theatre of the time, the Globe, and he easily won the favor of King James I. That alone would have been enough to write his name in the books.
This question is great one to ask. I often tell my students that rather than whining about what they "have" to read, sit back and take in a few lines to see just how masterful he really was with the language. It's quite impressive.
A fine question. The short answer is, greatness.
The somewhat longer answer is, Shakespeare was successful in his own time, but not the near-god he is now. He was one of several high profile playwrights in his own time, but not the only one. However, with time most of his contemporaries faded into historical figures, while Shakespeare remained alive, with new generations finding new ways to relate to him.
The general reasons he's considered this great are the characters he created, and the amazing poetry contained in his plays. Figures like Hamlet, Othello, Iago, Falstaff, and others stay alive in the mind long after they've left the stage, and the lines from his plays… amazing.
If you're interested in a more detailed discussion of this, the April 2007 issue of Harper's magazine has an entire article on this process. (I'll include the link, but it's only available online to subscribers.)
And Shakespeare's not the only writer to attain this stature. Homer was more influential in classical Greece, and has stayed famous for thousands of years, compared to mere hundreds for Shakespeare.