The main reason that certain individuals (against the overwhelming consensus of scholars in all related fields) claim that Shakespeare didn't write the works attributed to him is that we lack much in the way of biographical information about him. And why is this? It's because at that time--indeed, for most of human history--only the ruling elite were the subject of biography. (That includes critical biographies and outright hatchet-jobs as well as hagiographies.) Shakespeare was a commoner (albeit a moderately well-off one), and therefore didn't have scribes at his beck and call to record his comings and goings. Nor did he have a devoted secretary-friend like Boswell to do so.

Popular comics and fantasy author Neil Gaiman, interviewed in The Sandman Companion, suggests another reason for the anti-Stratfordian phenomenon. Some people, not necessarily consciously, have a bias towards the nobility being the only ones capable of producing literature of Shakespeare's quality. In fact, as Gaiman points out, most of the greatest authors (and other artists) in history were not aristocrats. Indeed some, like the plurality of bards we call "Homer," or the author(s) of Beowulf, were anonymous folks who barely eked out a living and who were neither formally educated nor even functionally literate. And yet no one finds it necessary to posit that it was Solon who wrote the Iliad or Alfred the Great who wrote Beowulf.

Honestly, I can't believe this is still the subject of debate. Why can't we let Shakespeare be Shakespeare?