William III was King of England because of his marriage to Queen Mary II of England, who shared power with him in a so-called "Crown Matrimonial." But he had earned this privilege by throwing the Dutch army on the side of the "rebels" against King James II of England (Mary's father).
William's real power base was as the Prince of Orange, known on the "other side" as the house of Nassau, having inherited this position as a descendant of William the Silent, the first "king" of the Netherlands. During the early 18th century, the Dutch considered themselves descended from the Batavians, a Roman-era people who had inhabited the Netherlands.
William added this adjective to his title to highlight his "Dutchness," because Nassau was actually a German place and title. Another way of reading his full title was as the "British-Dutch" German noble.