We don't.
William Shakespeare was born in 1564 and, at that time there were no official government records of births, marriages and deaths.
There were church records of baptisms, weddings and funerals and there is a record of Shakespeare's baptism on 26th April 1564.
On the assumption that the baptism took place within a few days of birth, Shakespeare's birthday is celebrated on 23rd April, which is the feast day of Saint George, Patron Saint of England.