The main reason that people have questioned whether William Shakespeare wrote his own plays has to do with his education. He was not a university man and most of the plays of the day were written by university men. His education went through the age of 12 (traditionally), but there is newer proof that he continued his education to the age of 14, 15, or even 16. What we fail to take into account is that he essentially had a Classics education and that even by the age of 12 he was fluent in Latin and had some knowledge of Greek. What this means is that he was educated in the very areas that figure quite strongly in his choice of subjects. But he was not a university scholar and many have been unable to wrap their minds around the idea that his best work was as brilliant and iconic as it was, or that his influence has continued into the present time. And so many people have come up with theories about who Shakespeare really was and they have presented a great deal of "evidence" so called, to back up their theories. These theories get awfully involved, however, because they are just that, theories.
Many new facts are coming to light as the old Elizabeth records are looked into, but there are some very simple common sense facts that would lead most of us to believe that Will Shakespeare was the greatest English writer ever.
Probably the most telling thing is that he existed within the world of the Elizabethan theatre and this was not the huge Hollywood concern we have today, with actors working out of all sorts of countries, and more people than can be named. There were not that many theatre companies around. And they all competed for the same audiences. And the great playwright, Ben Jonson, himself a university man, accepted W.S. as W.S. within his own time. Given the animosity towards W.S., if it could have been proven that he was not the writer, someone would have gone after him on those grounds alone. Instead, there were some who tried to put him down for flying too high, but no one questioned his authorship during his lifetime or even in the years afterwards, when the first portfolios were brought out. Ben Jonson wrote a moving and telling tribute to Will on his death, though they had been great rival playwrights in life.
I don't think we moderns realize the difference between the amount of information that was available in those times and the amount we are inundated with today. The information available in those times was much less. People had time to think and to reflect and Will had the framework (Classics plus real life living on the ground knowledge, the persecution on religious grounds of his own relatives--at least these for certain)which would let a man of natural genius flourish as a playwright. In other words, though he wasn't a university man, he was a man of his own world, living at the center of that world.
His work was carried on as a member of a company which was both established on site and traveled extensively. As did all the players he did a lot more than just act. His plays were enacted by his own company. When traveling they lived closely. Will would have have made corrections while the play was being produced. In short he carried on his work in an extremely public way. It is very difficult to imagine anyone getting away with producing someone else's plays in that kind of setting.
It has been argued that Will would not have had the knowledge of places and names to write his works. But London was a world capital and a port city. People would have come through on all kinds of business, talked of all kinds of places they had been, all kinds of things they had done. Will was a borrower of knowledge in his plays. He took old stories and made them new. What he displayed in his best works was a deep psychological and spiritual understanding of the personality and the soul of man. That is not something a university education can give. He effected such a wonderful balance between words and substance, a balance rarely achieved in literature, even good literature, some cannot forgive him for it and believe he is the great writer.
Another reason authorship has become a "mystery" is that Will's originals were all destroyed by fire. So there is no original copy of a Shakespeare play. And people love a mystery. It is only recently that more and more Elizabethan law records are coming to light which give more credence to Shakespeare the man. And learning about Shakespeare the man helps us in validating Shakespeare the writer.
The best way to determine your own opinion, however, is to read widely on the life of Shakespeare. Don't just read the Shakespeare was X theorists, however. Read the modern standard works on his life. Search out a video series which is called, if I remember correctly, "In Search of Shakespeare". It is an English series. Make up your own mind, but gather enough information to make a good determination and come at it with an open mind.