Why did English change so much between Chaucer and Shakespeare?

The printing press changed everything.

Prior to Gutenberg, English was primarily a spoken language and stories were often passed on in the oral tradition. The introduction of printed works in the mid-15th century had two major effects:

1. Standardization, as printed works were distributed beyond the reach of the local authors. This led to standardization of the language from what had been myriad local dialects to a more common single lexicon.
2. Stabilization, as the shift from an oral tradition for recording stories to a printed one fixed the form of the tales being written. As you can imagine, passing a story verbally for a century leads to many slight alterations of the story and even of the language in which the story is told. Once printed, however, its form is essentially fixed and will change very little - and the same is true of its language.

It is this second effect which results in what you've noticed: prior to the mid-15th century, the English language changed much more rapidly than afterwards.

Example of some research into this effect: http://homes.chass.utoronto.ca/~cpercy/courses/6361dickens.htm

I think there are two major factors at work.

1) Chaucer was the first person to write and publish in what was Middle English (check your copy of Canterbury Tales, I bet it is a translation into "Modern English" although the English in my copy isn't very modern and is still very difficult). Before then back to the time of the Norman Conquest the main language of the educated / ruling class was the French of the conquerors and so secular published material was either French or Latin.

Once Chaucer published everything was up for grabs and change was fairly rapid. Books were rare and normally one upper class person read a book out loud to his/her less educated peers. Chaucer's work extended the written word to the middle classes if not the proletariat. His bawdy tales in the vernacular naturally attracted a much wider audience and perhaps marked the start of literacy beyond just the elite in England.

2) Society went through enormous changes between 1400 and 1600. Elizabeth turned England into a world power. Having said that Spencer's "Faerie Queen" 50 years earlier may mark the start of modern English and that was before Elizabeth's time. These societal changes were reflected in the language.

The European development of the printing press in about 1450 was important but I don't see how that accounts for the differences. After all it comes only 50 years into the 200 year period. Note that the Chinese had the printing press much earlier.