See Part One and Part Two, if you're interested.
Why should you, or I, or NPR care about who wrote the texts attributed to Shakespeare?
There's that whole thing about the truth, and how important that is. But we're all going to say we're invested in finding the truth, and I'll assume that the anti-Stratfordians are as interested in finding the truth as I am.
What's at stake? Well, for some of us, there are career implications: if I find Love's Labors Won, I win. I might just be able to parley that find, and writing about it, into a position in a better geographic location. At the least, I'll get my name in the papers and maybe sell a book. So there's the personal profit motive: Shakespeare is big cultural capital. Most academics are pretty convinced that Shakespeare wrote Shakespeare, perhaps in part because we've set our careers on that? Maybe. But most Shakespeareans I know really haven't: we don't do much biographical criticism, don't try to psychoanalyze Shakespeare through his works, or talk about what he really thought by mining quotations from characters.
And when it comes to teaching Shakespeare, the texts are pretty challenging and exciting, and so I rarely talk much about the authorship question. To me, it would be sort of like an intro astronomy class spending time on Greco-Roman mythological explanations of the planets. Time is always an issue; it's one of the greatest limiting resources in academic studies, and I usually spend mine in ways that I enjoy way more than worrying about authorship. There's also the problem that for any potential author you demonstrate probably didn't write the plays, an anti-Stratfordian can find another early modern person that you'd have to read up on. It feels endless.
And to be honest, I've read a lot of early modern letters and such, and most are historically and culturally interesting, but not nearly as fun as a play. Plays are tremendously rewarding intellectually for the effort and time put into reading them. (That's one of the reasons I'm a lit person instead of a history person; history people like letters and parish records the way I like plays. And I'm glad they do, because I learn lots from them!)
It's worth asking what would convince someone to change their mind about something.
What would convince me to change my mind about the Shakespeare authorship question? I think early modern evidence would be necessary, either something to show that William Shakespeare didn't write the texts, or something to show that someone else definitely did. Manuscripts of one or more of the texts with good provenance information, showing that they date before the printing and are in a hand not otherwise associated with Shakespeare (a manuscript in the hand of Ralph Crane, for example, wouldn't convince me of his authorship or anyone else's). Maybe letters showing up that gave evidence of someone else's authorship? A good collection of manuscript sonnets, with good provenance, showing a different attribution.
Circumstantial evidence for another author isn't convincing at this point; what there is, attributions and such, point to William Shakespeare more than anything else.
With all the talk about authorship, it's worth noting that theater texts are pretty complicated, especially in the early modern period. (Sisyphus notes this in her comment on the previous post.)
The playwright (or a couple playwrights) wrote a play, and sold it to a company. The company made changes; actors made changes. They cut stuff, added in bits from other plays, and so forth. If they wanted revisions, they might get another playwright to make them, or to help.
They licensed the book of the play; in the process, the Master of Revels might make some changes, especially cutting something he didn't like. Some years later, the company might make pretty substantial changes to a play before putting it back into their repertoire for a while.
And then, if modern scholars are lucky, the book got into print. In Shakespeare's case, some plays get into print as very different texts (Hamlet, Lear, for example) fairly early on. Some don't get into print until 1623 (Shakespeare died in 1616), so he didn't have anything to do with printing the First Folio. (And there's another early modern attestation to Shakespeare's authorship: Heminges and Condell were actors in the company, and they attribute the plays in F1 to Shakespeare, as do the dedicatory materials.)
So worrying about Shakespeare as a sort of sole author isn't that useful. Rather, it's more useful to look at the texts, think about how they work, their contexts and meanings, and so forth. And way more fun, too.
Think about it: in the time it took you to read these three posts, you could have read Act 1 of Titus or something!