Shakespeare died at his home in Stratford, England. He was 52 years old. The exact date of his death is unknown, but he was buried on April 25th, 1616, and so it is commonly assumed he died on April 23rd, his birthday. The last revisions to his will were signed by him only a month before this, on March 25, and some scholars claim his handwriting and the fact that the will was never completely recopied into a "clean" copy indicate he was already seriously ill at that time.
Experts have suggested at least two tentative causes for his death. The first is typhoid fever. There is a late story that Shakespeare died of a fever after a night of carousing with friends. It is generally looked on with scepticism, but on the other hand there was a typhoid epidemic around the time of his death, so perhaps he did pass of a fever caused by illness, not drink. The second is cerebral hemorrhage or apoplexy. This is argued from his high-stress and intellectual lifestyle, some features of his portrait that appear to indicate arterial problems, and the hurried completion of his will.
Both of the above theories remain speculative, and in fact we will probably never know why Shakespeare died. In the context of his time, however, he had led a full life, and it is unlikely anyone felt that his passing was unusually early, however much it may have been mourned.