This page has a pretty thorough commentary on the topic.

To summarize, we have numerous contemporary references to Shakespeare as a writer of plays and poems, such as Palladis Tamia: Wits Treasury by Francis Meres, published in 1598. Meres attributes 12 plays to William Shakespeare, including "[Two] Gentlemen of Verona," "[Comedy of] Errors," "Love labors wonne," and "King John."

In 1609, a book titled Shake-speares Sonnets was published by Thomas Thorpe. The book is believed to be unauthorized, but definitely attributes the poems within to William Shakespeare.

One concern of a historic nature is to confirm that the William Shakespeare known as an actor in London is the same William Shakespeare of Stratford-upon-Avon -- after all, they could be two entirely unrelated men with the same name.

Thankfully, we have documents that tie them together, such as William Shakespeare of Stratford-upon-Avon's will, which bequeaths an amount of money to John Heminges, Richard Burbage and Henry Cundell; we have other documents that list William Shakespeare the actor performing alongside Heminges, Burbage and Cundell in The King's Men acting company.

So we know that Shakespeare the actor was Shakespeare from Stratford-upon-Avon, but we also know that William Shakespeare was a writer.

In 1601, the students at Cambridge put on a play called The Second Part of the Return from Parnassus. One line from the play establishes William Shakespeare as a writer:

Few of the vniuersity [men] pen plaies well, they smell too much of that writer Otiid, and that writer Meta morphosis, and talke too much of Proserpina & luppiter. Why heres our fellow Shakespeare puts them all downe, I and Ben lonson too. O that Ben lonson is a pestilent fellow, he brought vp Horace giuing the Poets a pill, but our fellow Shakespeare hath giuen him a purge that made him beray his credit

This line establishes Shakespeare as a contemporary and rival of Ben Jonson, another playwright of the era (and also seemingly noting that Shakespeare is a favorite of the students at Cambridge)

After Shakespeare's death, the same Ben Jonson wrote a poem "To the memory of my beloved, The Author Mr. William Shakespeare," in which he says,

Sweet Swan of Avon! what a sight it were

To see thee in our waters yet appeare,

And make those flights upon the bankes of Thames,

That so did take Eliza, and our James!

The poem appears in collection of Shakespeare's works, First Folio of 1623, published by John Heminges and Henry Condell.

I've just highlighted and expanded on a few of the points listed in the first link. In short, we have a lot of contemporary examples that confirm that an actor and writer named William Shakespeare lived and worked in the time and place we expect him to have worked. Given that confirmation, we have little reason to suspect the works we attribute to William Shakespeare aren't the work of this author and actor.

While it's possible that it was a pseudonym, it's highly unlikely. The people of William Shakespeare's time definitely knew him, interacted with him frequently, and none of them put forth any suspicion that he was a fraud or not the writer they believed him to be.

hi! there's always room for more discussion, but there have been a few threads on this. catch up on previous responses here

Is there any solid evidence that Shakespeare's works were written by others?

What are some of the arguments against Shakespeare's authorship?

Do you think Shakespeare actually wrote the famous plays? What do the experts think on the conspiracy?

And how do we know that they weren't written by more than one person?

Ironically, one of the most satisfying pieces of evidence against the 'disguised noble' hypothesis is the fact that Shakespeare did collaborate with a number of other playwrights - as was utterly typical for theatrical practice in the period.

Some plays are established without much doubt as being co-authored: Two Noble Kinsmen and Henry VIII with his successor John Fletcher, Timon of Athens with Thomas Middleton [both of whom are substantial figures]. Beyond this, there's a reasonable amount of evidence that other plays (Titus and Pericles get mentioned a lot) have some non-Shakespearean material, though whether that represents revision, co-authorship, or some other arrangement is very difficult to know.

There's a growing movement in Shakespeare scholarship to look elsewhere for possible revisions or additions by Shakespeare to older plays (again, a well documented practice), and there are now tentative assignations of authorship for a range of other material. Controversial stuff, but the point is this: we've got a very good bulk of material that suggests that Shakespeare was intimately involved in the theatrical moment of his time, and ironically, evidence that he didn't write all of his material alone is both utterly typical for the period, and a difficult piece of evidence for non-Shakespearean hypothesis to deal with.