It would be interesting to know, though impossible to find out, to what extent those who claim Shakespeare didn't write his own plays are motivated by jealousy. Many writers find it difficult not to be jealous of ol' Billy—he's the one writer nearly universally touted as the greatest who ever lived. Most, however, simply rejoice that such pure literary magic was distilled for subsequent generations, and that by an alchemist of less-than-noble origins and somewhat obscure life.
The idea that Shakespeare didn't write Shakespeare wasn't born with the movie Anonymous. So-called scholars and critics point out that the one called William Shakespeare sure knew a lot about the world for one who'd never left England, or that he was pretty well acquainted with the ways of the English Court for a commoner, or that an uneducated bumpkin from Stratford-upon-Avon could never have produced a coherent body of work that transcends the volumes of far more academic writers.
Anyway, he ended his days as a businessman back in his hometown, away from the chaos and mud and spectacle of London. Is that really how the greatest writer who ever lived would end his days? And what about the mystery of his handwriting? was he really illiterate? if he could read and write, wouldn't he have left behind more than a few signatures and some misspelled words? At first glance, the evidence seems compelling to attribute "Shakespeare's" work elsewhere, to the Earl of Oxford, perhaps, or Sir Francis Bacon, or some cadre of contemporary playwrights.
In the end, the reasons for finding another author for Shakespeare are more related to worldview than personal motives. Postmodernists like to obscure authorship in an effort to show that standards and absolutes are simply human constructs; modernists (and postmodernists) like to impose contemporary values and ideas on peoples of the past. The latter is probably the biggest motivating factor in the Shakespeare debate: the Elizabethans had no concept of ownership or copyright when it came to texts, but most higher criticism supposes they did.
Ultimately, there's no substantial reason to believe William Shakespeare was not the source of the plays published in his name. What is absolutely certain is that no one before or since has so thoroughly and completely captured the scope and essence of the human predicament, or done so half as beautifully as the Bard of Avon. His poetry is somewhere between biblical oracle and Miltonian pedantry; his prose the most poetic lines penned by a seller of wheat (or a nobleman, or anyone else for that matter).
All good literary education begins with Shakespeare. A lot of it ends there, too. The reason so much modern fiction is unreadable is because writers don't look far enough back for inspiration; a few generations ago, writers read Homer, Milton, Shakespeare and Dickens. A noble company, ironically helmed by a commoner with less education than those who call themselves scholars in his name. We carry most of his plays, and a variety of resources to help you understand them. Just remember—Shakespeare's plots are borrowed and recycled; it's the genius of expression that sets his work apart.