Is it merely because of how prolific his writing was? Was his writing better preserved, so that other, earlier works using the same words were lost? Or is there really some aspect of Shakespearean plays that make them more likely to utilise words not yet recorded in writing?
It's because he was obsessed with words at a time when the spelling and usage of the English language had yet to be codified.
The first English dictionary, Samuel Johnson's A Dictionary of the English Language, was first published in 1755 (almost a century and a half after the Bard's career) and it cites Shakespeare more often than any other author as the origin of terms and phrases. Shakespeare therefore sets the precedent for the definitions and spellings of more than 1,700 English words.
Until the 14th Century, French was the official language of England due to Norman/French cultural dominance. English was not recognized as the official language until 1509 and up until 1583, the rhetoric of what would later become Modern English was primarily indebted to Chaucer. Other than Chaucer, the relative lack of written records from the period makes the innovation of our contemporary common tongue uncertain. According to literary critic Boris Ford:
Before the arrival of Shakespeare to London, there was little hope for the future of English but by 1613, when Shakespeare's last work was written, the literature of modern English was already rich in varied achievements, self confident and mature.
The scope of Shakespeare's vocabulary is simply mind-boggling. The King James Version of the Old Testament contains 10,867 unique words. The works of Milton use ~15,000 - 16,000. According to the Shakespeare Lexicon and Quotation Dictionary, The Bard employs an astonishing 27,780 unique words.
Some of these new words were his invention, others were simply written down for the first time by him. Alas, it is difficult (perhaps even impossible) to ever tell which is which.