How do people know he did it?
Is there any proof that Shakespeare wrote his plays?
Well, if you could find rock hard solid evidence that William Shakespeare wrote his plays, that would eliminate all doubts, and make you very famous indeed as this is an issue that has been debated since the 1800s, with the jury still out deliberating the matter.
On my blog I have published both sides of this question in extensive detail in the two entries entitled British Literature: In Defense of William Shakespeare, and British Literature: The Case Against William Shakespeare.
Hope these will help you understand what is known as the Anti-Stratfordian Theory, or those who believe Shakespeare did not write these plays he is credited with having written, and the side that popularly believes he did.
Alex Knapp of Forbes magazine wrote an excellent article on this subject. See: http://www.forbes.com/sites/alexknapp/2011/10/19/yes-shakespeare-really-did-write-shakespeare/
Shakespeare was both an author and actor, and he would have modified his works to improve actors' performances. He would also have modified his plays in response to his fellow actors' and playwrights' critiques of his works. He was also apparently critiqued in pamphlets of his day. See the above article. While he may not have authored every word, it appears that he wrote many of his plays.
Matthew, you've made an interesting point for transition which I would like to take. This is an extremely interesting question when you look at it through the lens of Stephen Greenblatt, Roland Barthes, and Michel Foucault.
Stephen Greenblatt would be most interested in reading Shakespeare because of what "Shakespeare the man" accomplished and achieved in his life juxtaposed next to what he likely wrote and why he may have written it. Therefore, he would care the most about an answer to this initial question, as one can watch in his PBS interview about his book entitled, "Will in the World: How Shakespeare Became Shakespeare."
Roland Barthes in "Death of the Author" on the other hand begins with your question: does it matter? He responds with something along the lines of "studying the author himself has nothing to do with what he wrote." He calls to metaphorically "kill the author" in literary criticism.
In "What Is an Author?" which is a response to Barthes, Michel Foucault argues that yes, it would be awesome to be able to separate the author as he lived from the his texts that he left behind; however, that is impossible. In terms of Shakespeare for Foucault, you can kill William Shakespeare, the actual man that lived and probably wrote plays, but you cannot kill the function of author, or what Shakespeare represents in his texts, because not only do his texts outlive his bodily form, but they continuously reinforce a reader's impressions and schema of the word "Shakespeare."
Ultimately, this simple question, "Is there any proof that Shakespeare wrote his plays?" is reminiscent of decades of theoretical discussions.
When one wishes to make exceptional claims one must deliver exceptional proof. If one wishes to theorize that Hannibal's army used explosives and chemical weapons (there is an actual academic article that proposes this rather eccentric theory) then the author must find chemical weapons laboratories that existed in Carthage at the time of the Punic Wars or some other solid proof, otherwise the theory can be discounted. Similarly, if one wishes to prove that Shakespeare was not a great author then one must offer solid proof.
Further, what is often offered as proof is easily explicable.
1: He spelled his name different ways. Explanation: First, accurate spelling was not considered important in the Elizabethan Era. It did not become at all important until the publication of the first dictionary, which was not until the Regency period. Second, people still do that today. John Mellencamp, Prince (The Artist), and "Puff Daddy" have changed their names several times. In the case of my family, think of the various spellings of "Osbourne" "Osborne" "Osbourn" "Osborn" etc. Each is a variant of the same name, and each was chosen as a preferred spelling by some ancestor of each line of the family
2: Some of his stories were better than others. Explanation: Think of music. Let's apply this theory to the Beatles. Was every Beatles song of equal quality? Did every album receive identical reviews? Of course not. Some (especially those written under the influence of Yoko Ono were awful) others were classics. Think of your favorite band? Are all of their albums equally good? Do you like every song equally?
3: He was uneducated. Education was not as formalized, indeed it is still not completely formalized. Simply working in the theatre business would have exposed him to what would amount to a liberal arts education today. As another example, Poe was a better poet at age 12 than most people are after years of training and classes.
So, until there is a box of lost Shakespeare manuscripts (say Love's Labors Found) uncovered in the Lord of Oxford's manor, we have to accept Shakespeare as the author and the idea that he was a false front for another as a theory with no firm evidence.
Is there proof you wrote this question on Wyzant?
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.
The most profound answer to the question is (in my humble opinion): Does it matter? The scholarly detective work is interesting, to a point, but I believe that time is better spent studying the plays and poetry. The truth is, there are certain types of people who want their names attached to the "Greats" of history. In some small way they become Shakespeare by working on this problem, or by becoming an expert on "Hamlet" for example. They become Einstein by writing about Einstein, a Beatle by writing about The Beatles. Have you ever felt that pull? I have. Let me make it clear that I don't for a second believe all scholars pursuing this issue are like that, but there are some. There have been, are, and will be, many great Shakespearean scholars. I would love to listen to them discuss the Bard, wouldn't you?