There are a number of reasons why some people believe that Shakespeare did not write his own works. One reason is the lack of evidence to support Shakespeare's authorship. There is no known manuscript in Shakespeare's handwriting, and there are no contemporary accounts of Shakespeare working as a writer. Additionally, some people argue that Shakespeare's works are too complex and sophisticated to have been written by a man with only a grammar school education.
Another reason why some people believe that Shakespeare did not write his own works is the fact that there are a number of inconsistencies and errors in his works. For example, in the play Julius Caesar, the character of Brutus is referred to as "Brutus Cassius" in one scene. Additionally, some of Shakespeare's plays contain anachronisms, or references to events that had not yet happened when the play was written.
Despite the evidence that suggests that Shakespeare may not have written his own works, there is also evidence that supports Shakespeare's authorship. For example, there is a number of contemporary references to Shakespeare as a playwright, and his plays were published with his name on them. Additionally, Shakespeare's works are full of personal details and insights that seem to be consistent with what is known about Shakespeare's life.
Ultimately, the question of who wrote Shakespeare's works is a matter of opinion. There is no definitive evidence that proves or disproves Shakespeare's authorship, and it is likely that the debate will continue for many years to come.