I think there are two reasons: the first is emotional, the second is slightly more cynical.
Shakespeare was writing at a time when the English language was just starting to expand and become as expressive and responsive as Latin, which until then had been the language of so-called high culture. Language, and therefore thought, was undergoing immense change; Shakespeare was a crucial part of that.
He made language do things for the first time, and showed how it can be simple, reflective, philosophical, lyrical – or all at the same time. His extraordinary creative energy and verve created plays that were deeply understandable, both about human experience and also massive national upheaval, in a native language that had rarely dealt with such ambitious topics on the stage. He raised questions about the nature of love, death, power and leadership – themes that remain eternally relevant.
More cynically, various historically contingent factors made Shakespeare our national author. English national identity was being consolidated at the time Shakespeare was writing, and a national author was felt to be an important contribution to that notion of identity. While during Shakespeare’s lifetime, the national author was thought to be Ben Jonson, that all began to change in the mid-seventeenth century.
In the run-up to Civil War, the theatres were closed in 1642, and only reopened in 1660. That meant there were 18 years where people could only read plays, which then placed an emphasis on people who were published in books. Shakespeare was one of those. This meant that people started to appreciate him as literature as well as performance. Then, when the theatres did reopen, we didn’t have any fresh playwrights, so we looked back to previous authors to out them on: Shakespeare was one of the first people turned to.
When he was performed in the 1660s, it was the first time we had women on the stage. Previously, all female parts had been played by young boys. Because Shakespeare has so many scenes of cross-dressing or undressing, for the first time we were seeing women taking off their skirts and showing their legs – so he had a boom in popularity for porn reasons as much as anything else.
In part because of that titillation, and the general enthusiasm for the return of the theatres, a certain cult of celebrity started to form around prominent actors and actresses, which then attached itself to Shakespeare. His death on St George’s day, and rumours that he was born on that day too, also helped create the myth of him as the English author.
As a producer of great literature, great performance, and in a curious way, great porn, Shakespeare became a canonical figure in the English literary landscape.