Much of the dating of Shakespeare's plays comes from information in the Stationer's Register of London. The Stationer's Company was a trade guild granted the right to maintain a list of all printed materials in London. So, if you wanted to print your poetry, plays, other writings, have a book bound, etc., you'd have to pay a fee to have it listed in the Stationer's Register. Many of Shakespeare's plays are registered with the Stationer's Company. So it's an invaluable resource for understanding when Shakespeare (or members of his company) had his plays published. While the publishing date is not the same as the authorial date or the performance date, it does give historians a picture into the order in which Shakespeare likely wrote his plays; thereby, also providing a narrative for Shakespeare's artistic endeavors over time. It also gives insight into potential contextual clues that inform the intents behind the plays he wrote--especially since Shakespeare's career spans two different monarchs of England. Furthermore, Shakespeare was attached to several theatres in Elizabethan England--namely, the Theatre (1576), the Globe (1599), and the Blackfriars (1608). The Theatre and the Globe are both open-air theatres similar in style and construction. But the Blackfriars was an indoor space that was converted into a theatre (using a layout that predates the pit, box, and gallery arrangement but is nevertheless quite similar). So, knowing when a play may have been written helps us understand the intent behind its writing according to current events of the time; the space and audience (since the Blackfriars was a venue accessible to those able to afford a more expensive ticket) he imagined for its performance; and when the play was published in relation to its performance.
Allow me to provide a quick case study to show how all this can come together to better inform one of his most well-known and enduring plays: Hamlet. The Stationer's Register entry for Hamlet is dated July 26, 1602. But, historians mostly agree that the play was written and performed before this date, and probably closer to 1600. Because the Stationer's Register lists it in 1602, we can be very certain that the latest date for a first performance of Hamlet would be in that year. Because we know that the Globe was constructed in 1599 and Shakespeare (and others) didn't take over use of the Blackfriars until 1608, we can also be very certain that the first performance took place at the Globe Theatre in Southwark. We can also be certain about who performed what roles, with Richard Burbage the likely candidate for the titular Hamlet because he was a prominent member, and leading actor, of the Lord Chamberlain's Men (Shakespeare was a member as well) in and around the year 1600. The c. 1602 dating also helps frame the historical context from which the play originates. It was written in the last years of Elizabeth I's life and rule, during a time of English ascendency across Great Britain and the European continent; allowing English Protestantism to take deeper root across the country as the state-established religion--perhaps informing some of Shakespeare's choices about where the play is set, the time period in which it's set, and the mention of places of study such as Wittenberg, etc. But that's beyond the scope for my answer.
BUT! (And this is the fun part.) 1602 is not the only date associated with Hamlet. The very next year, in 1603, there is a register for publication of the play by Nicholas Ling and John Trundle in a format that has come to be known as the "First Quarto" (a quarto was, essentially, the size of the document upon which the play was printed). So Ling and Trundle published Hamlet in a quarto, but if you read this edition, you'll find that it has striking and numerous differences from the play that we're all familiar with today. So where did the edition of the play that we read in high school come from? It wasn't published until the First Folio edition of 1623 (the famous one compiled and published by Shakespeare's fellow actors John Heminges and Henry Condell after his death). Furthermore, James Roberts (the person who originally registered the play in 1602) had a second quarto edition printed in 1604. All of these editions of the play have recognizable variations. So, which one are we to believe is more authoritative and/or representative of what was performed at the Globe? Well, the best answer is that we don't know. But, all of these editions of Hamlet do tell us one thing for sure: the play was very popular in its own time. And it's likely that each publication represents an attempt to capitalize on the its popularity because the Stationer's Register is essentially an early attempt to provide documentation surrounding the publication rights to a literary work (a lot of the history of today's copyright laws have roots in this period of English history through the work of the Stationer's Guild). We don't know much, if anything, about the individuals that published the early quartos and whether they had any connection to Shakespeare. But we do know that Heminges and Condell were Shakespeare's peers and colleagues who compiled his plays in as authentic a way as they could. So, that's why historians and literary critics tend to view the First Folio as the likely authentic representation of what Shakespeare wrote and what was performed.
So, to wrap this up, knowing the dates for Shakespeare's plays helps us provide a more complete picture of the world in which Shakespeare was writing as his career spans about 20 years. It also helps us put together a picture of Elizabethan and early Jacobean London that tells us something about the cultural place of theatre for the city and England at large.
Sources:
- Digitized Stationer's Register Entry for Hamlet (with bibliography): https://shakespearedocumented.folger.edu/exhibition/document/stationers-register-entry-hamlet
- Richard Corum, Understanding Hamlet: A Student Casebook to Issues, Sources, and Historical Documents, 1998.
- Paul Menzer, The Hamlets: Cues, Qs, and Remembered Texts, 2008
- Zachary Lesser, Hamlet after Q1: An Uncanny History of the Shakespearean Text, 2015
EDIT: You can find the differences between the Hamlet publications in the famous "To be, or not to be" speech here: https://en.wikipedia.org/wiki/To_be,_or_not_to_be