This is a great question. There are many thing that we can say. So, let me limit myself to a few points.
First, Shakespeare has a profound effect on the English language. For example, some of his words and phrases are still apart of our world. We can say this his English has transformed the way people speak in the past and we still feel his influence.
Second, he has influenced our imaginations with his literary works. Just the fact that we read his works today say that there is an profound influence. We see this also in many remakes of Shakespeare in the theater and in movies. For example, the novel, Brave New World by Aldous Huxley use the words of Shakespeare (The Tempest, 5.1). In addition, Shakespeare has also influenced the arts and music.