A perfect answer to this question is impossible, as we cannot know with certainty what motivated him in the 1590s to be in London and to become a playwright. However, a few historical contexts point to plausible influences.

First, because his father was a glove maker and trader of agricultural products, Will would have qualified to attend the Stratford grammar school. This school was considered a strong educator of young boys among many strong grammar schools founded by Henry VIII after he dissolved the monastic schools (Christopher Marlowe's school in Canterbury is also considered strong). The curriculum here would have been the medieval trivium, a solid training in grammar, logic, and rhetoric. Boys were trained to read and to speak, and the method of training in the classical mode produced highly literate and articulate students. A feature of this pedagogy was writing imitations, "speech in character work," and modest theatrical performance to train for work in the law courts. It is not at all surprising that a generation of young men would come out of these schools and produce a rich body of literature and theater coinciding with the creation of fixed theaters for stage plays. Shakespeare would likely have seen popular playacting when traveling companies passed through Stratford when he was growing up.

The first successful theaters built in London went up in the late 1570s. We don't know what Shakespeare was doing during a few years of his young adulthood, until he shows up in London in 1592. Two plausible answers would be that he had moved to London and was using his education by working in theater, perhaps as an apprentice actor, or that he was traveling with a theater troupe learning his future trade. When he did find himself in London and the record begins to document his life again, he is lambasted as a playwright whose growing popularity was surpassing that of the university-educated writers.

So, why did he write plays? He had the ability and the market to do so. It was a hot new industry and while money was tight for actors and even playwrights, becoming a stockholder in a company like Lord Chamberlain's Men and owning a share in one's own theater, like the Globe, made it a lucrative profession for a commoner at the time. We know that travelers to London commented on the popularity of theater and then, like now, a person visiting the city would want to take in a play, given that the English capital was a center of activity for this increasingly popular art form.

Shakespeare never answered this question directly, but we can easily guess several motivations for him to write his plays.

The first is a simple desire to please other people, to do something remarkable that will attract praise. Shakespeare comes as close as he ever did to a direct statement in the final lines of The Tempest, probably his last full play, where Prospero seems to step partially out of his stage role and begin speaking in a different voice:

Gentle breath of yours my sails
Must fill, or else my project fails,
Which was to please. (The Tempest, epilogue)

Prospero, the character, has not been motivated by a desire "to please" -- this is Shakespeare talking here.

A second motivation was probably a desire to comment indirectly on contemporary events, to offer praise or warnings. Julius Caesar, for instance, is clearly a sermon on how it is permissible to deal with tyranny, and Macbeth both flatters King James and warns him of the results of ambition taken too far. The England of his time was not a democracy, and such indirect commentary gave Shakespeare a much more powerful voice than he otherwise would have had.

But finally, and probably most important, Shakespeare wrote plays to make money. He came from a family of modest means, living away from London, and made himself into a "gentleman," a successful enterpreneur, and a considerable property owner, all through the power of his pen. Thus, despite the genius of his work, we need not doubt that the primary motivation for Shakespeare's plays was a common one -- to put food on the table for himself and his family.