Shakespeare's death, April 23, 1616, in Stratford-upon-Avon, is considered a mystery because the exact cause of his death is unknown. We actually consider there to be three possible causes.
The first hypothesized cause we have comes from an entry the vicar of Holy Trinity Church in Stratford, John Ward, made in his diary 50 years after Shakespeare's death. Ward was a fan of Shakespeare's and recorded that after having a "merry meeting" one night with Drayton and Ben Jonson in which Shakespeare "drank too hard," Shakespeare then "died of a fever there contracted" (Mabillard, "How Did Shakespeare Die?"). In other words, according to Vicar Ward, Shakespeare died of a fever after drinking too much. However, historians believe that the account may not have much bearing. The one thing that does hold true about Ward's account though is that Shakespeare may have indeed died of a fever, as it is known that in 1616 there was a typhus epidemic, which is a disease caused by a parasitic bacterium. Therefore, the first hypothesis we have is that he died of a fever after drinking too much, while the second hypothesis is that he simply died of a "fever," otherwise known as an illness called typhus.
The third hypothesis we have comes from an account of Shakespeare's physician Dr. John Hall, and was recorded in C. Martin Mitchell's biography of Shakespeare. Dr. Hall actually speculated that Shakespeare may have died of a "cerebral hemorrhage or apoplexy" (Mabillard, "How Did Shakespeare Die?"). Dr. Hall made this speculation based off of Shakespeare's Droeshout, an engraved portrait of Shakespeare that was published in his collected works in 1623. Dr. Hall noticed some shadings on the portrait that would indicate "thickening of the left temporal artery--a sign of atheroma and aterio-sclerosis" (Mabillard, "How Did Shakespeare Die?").
Hence, Shakespeare's death remains a mystery because we do not have an accurate recording of the cause of death. Instead, we have three different hypotheses.
Another interesting note to add: Shakespeare is believed to have written his own epitaph, which can still be seen on his grave today if you care to visit it,
"Good friend, for Jesus' sake forbear, | To dig the dust enclosed here. | Blessed be the man that spares these stones, | And cursed be he that moves my bones."
This has been respected, as far as we know, and we're assured that those involved with Holy Trinity Church's restoration in 2008 were particular careful around Shakespeare's resting place to avoid the curse!