The simple answer is that there’s a written record. And the O.E.D. is a brilliant document that shows the first written use of a word and which document it comes from. There are a few instances of other writers creating words but Shakespeare was better at it. He was also a relatively popular playwright so that may have helped his words gain traction. Remember, this was a moment when over about 60 years the English language was going through a huge shift that allowed new words in a way that had never happened before or since. Shakespeare wrote in the middle of that shift. Finally, Shakespeare’s impact was finalized by the lifting of Shakespeare in England in the centuries following his death, and in America as a mark of culture. As people settled the West, they would often have two books: the Bible and the works of Shakespeare. That cemented Shakespeare’s words in the language.