What are some facts that give evidence as to why Shakespeare is famous?

Probably no other writer of modern times—meaning since about the year 1500—has achieved the international recognition and appreciation accorded Shakespeare.
It would be too easy to say that the reason for this is simply the fact that Shakespeare is an incomparably great writer. Few of us would be satisfied with such a simplistic explanation. It would be more useful to look at his fame as increasing in stages, aided by historical events and processes independent of the inherent value of his work—though of course, if his work didn't have such incomparable value, these events wouldn't by themselves have caused his reputation to rise to the level at which it has been for many decades now.
It's one thing for a playwright's works to hold the stage even after his death, after tastes and artistic values generally have changed. It's another situation, however, when critics and intellectuals begin to pronounce a writer as the great genius of his time or as one who has transcended his time and become a cultural icon. The latter happened to Shakespeare in the century after his death. In the eighteenth-century, both Alexander Pope and Samuel Johnson produced critical editions of Shakespeare. Johnson's preface to his edition was especially path-breaking because, though he enumerated what he considered aesthetic flaws in Shakespeare's plays, he also expressed the view that the works were incomparably great and that there was almost something superhuman about their overall artistic merit, in the richness of their language and their incisive, realistic portrayal of human nature. This had not yet become a generally accepted opinion. Most English critics of the time (the 1760s) would still have considered a more elegant and "classical" writer, such as Joseph Addison, as the greatest English playwright. Johnson, though he admired Addison, stated that "Addison speaks the language of poets; Shakespeare, of men." In other words, those dramatists who had been praised by the critical establishment began, from the time of Johnson's evaluation, to be seen as stilted and as having failed to depict human emotion as honestly and directly as Shakespeare.
A few years after this, German writers and critics, such as Johann Wolfgang von Goethe and Johann Gottfried Herder, "discovered" Shakespeare. These men not only were anglophiles in general, but they wished to break free of the French models German writers, in an age when the Germans were creating a national literature of their own for the first time, had previously been encouraged to emulate. They saw Shakespeare, just as Johnson did, as more faithful to human nature, bolder, more emotional and more honest than the French classical playwrights such as Corneille and Racine. In the late eighteenth and early nineteenth-century, German translators, who were great poets in their own right, such as Christoph Martin Wieland, Ludwig Tieck, and Karl Schlegel, produced translations of Shakespeare's complete works. Shakespeare's fame thus began to spread across the continent, given the now European-wide reputation and authority especially of Goethe, who was in effect Shakespeare's chief "publicist" at this time, the period from 1780 to 1830.
What accounts for not just Johnson's evaluation, raising Shakespeare to a new level in the public consciousness, but for that of the German writers? Probably Enlightenment ideas and ideals are at the root. In the eighteenth-century, old notions concerning various areas of human thought, philosophy, religion, and aesthetics began to be questioned. Shakespeare, with the forcefulness and directness of emotion in his works, fit in with the Enlightenment conception of the world and of mankind. It wouldn't be an exaggeration to say that at this point a kind of snowballing effect occurred, eventually carrying Shakespeare's fame beyond the English-speaking world and Western Europe as a whole. The fact that Shakespeare came to be translated into so many languages and to be appreciated by foreigners, almost as if he were one of their own, belies the notion that poetry is somehow "untranslatable." One could even say that it's precisely because Shakespeare's verse is so rich and so striking that it can be translated so effectively.
Shakespeare's fame, as stated, grew by leaps and bounds as the nineteenth-century progressed. This was partly for philosophical reasons, as well as for aesthetic ones. A new sense of tragedy and pessimism, in some sense the opposite of the optimistic and self-congratulatory mindset of the eighteenth-century Enlightenment, became dominant in the Romantic era. Shakespeare's tragedies, though they had been written over 200 years earlier, were a fulfillment of the Romantic view of life, just as they had been, paradoxically, of the much, much different Enlightenment. And the same can be said of the 20th century and our own time, in which there have been, in the modernist and post-modernist periods, more attempts than ever before to show the negative side of human life and the world.
This brief sketch has outlined what I believe are the major events in Shakespeare's rise through history to the position he holds as an international cultural icon, as one whose reputation and artistic authority exceed those of all other writers of the past 500 years. There are many points that could be added. But despite history and changes in thought, the most important factor is still probably the intrinsic value of his works as poetry and as drama.