Shakespeare was real, there's quite a paper trail, he was incorporated, involved in court cases, bought houses etc. His existence is well established and beyond refute. That he wrote his own work is beyond reasonable refute. He was well recognized by other writers of the time, and famously criticized for immorality and pandering by some contemporary literary critics (men with a reason to out him if there was any doubt he actually wrote his plays). He was well known in his own day, and well known as distinctive (his "own honey-toned phrase", in the words of one author).
All conspiracy theories involve a generous portion of gullibility and insanity, but the conspiracy theories are especially pathetic re: this topic. They are mostly all based, IMO, on a desire to see history in a particular light. Those with a feminist bent would like it if the most famous author in English Lit were a woman....hence the Queen as WS (not enough to rule a country, I guess). Those with an elitist bent call for Bacon, because no random dude from a small town could possibly crank out some of the greatest prose and poetry in our language. It clearly took a genius like Bacon....except that it didn't, and there is not a shred of evidence. Edward de Vere (Earl of Oxford) has been put forward (only a nobleman could have done it....except de Vere was functionally illiterate, and could barely sign his own name). Johnson and Chapman have been suggested, but they were both successful writers in their own right, and had no need to set up such an elaborate ruse....plus, there is no evidence for either.
The only sense in which the theorists have a point is that Shakespeare never sold a book. All of his writings are cobbled together from his play scripts, notes and letters and heavily edited, and it all took place after his death. Nothing remains of Shakespeare's original work, only copies. So, in a sense, the real writers of Shakespeare's works are the legion of editors who pieced his (often conflicting) writings together into a cohesive whole.