The first point one should make is that there is absolutely no way to prove this claim. There have been many great and influential writers across a wide range of periods and cultures, and there is really no way to measure who is the "greatest." To say Shakespeare is somehow objectively "greater" than Homer, Basho, Lady Murasaki Shikibu, Dante, Sophocles, Tolstoy, or the anonymous author of the Epic of Sundiata is to make a claim that one cannot really support without being narrowly ethnocentric.
What one can actually claim is that Shakespeare is one of the most innovative and influential figures in English literature. He was important for the development of two genres, poetry and drama. In drama, he was a pioneer of the "mixed" drama which blended comic and tragic elements, often by the introduction of "rustics" or "mechanicals" into a drama otherwise dealing with the travails of noble characters. He also is generally credited with inventing the device of the quarreling lovers (such as Beatrice and Benedict in Much Ado About Nothing). Many of Shakespeare's phrases have become part of the common currency of the English language. He was also quite important for his contributions to the evolution of the English sonnet.