William Shakespeare wrote plays and poems, and in fact, it was one of his poems that helped him gain popularity: "As early as 1593 he had an overnight sensation with [Venus and Adonis]," a narrative poem (Michael Dobson, director of the Shakespeare Institute). There remains just one copy of the first edition, as the other copies were worn out from over-reading.
His work often spread through word of mouth, from the common people to those in higher standing, such as the Romantic poet Keats.
In Shakespeare's time, London was a tourist destination for theater, so his location allowed his plays to be seen by travelers. His work also spread to other countries through the influence of the British Empire:
Because of the British Empire, he’s translated into different languages and adapted by many different cultures. Germany had a special affection for Shakespeare and has always had that, I think. Some of Shakespeare’s own actors went to perform in Germany in his lifetime and it’s Germany where the first translation of Shakespeare occurs in the middle of the 18th century. Such is the extent of Shakespeare’s popularity, he becomes an honorary German national poet. (Paul Edmondson, Head of Research and Knowledge here at the Shakespeare Birthplace Trust)
Shakespeare's plays are frequently performed today, with directors often choosing to set the shows in different times and places. Scholars tell us this is not necessarily a new trend—the plays have been adapted and reworked for many years.
There was a period in history where the theatres were closed and then reopened in about 1660. At that time, Shakespeare’s plays were still being performed and were performed: although, were frequently amended by the next generation, sometimes quite hugely. There wasn’t at that time, any sense of reverence towards Shakespeare’s plays. They were nice blueprints that you could start with: if you wanted to add other characters, other scenes or alternate endings that was fine and up for grabs and you just did that. (Elizabeth Dollimore, Learning Manager at the Shakespeare Birthplace Trust)
There are a number of factors that led to the spread of Shakespeare's work, popularity, and eventual fame.