To elaborate, this reading of The Miller's Tale in Middle English is quite hard to understand as English. In contrast, the Shakespearean pronunciation in this Open University video on how Shakespeare's plays would have been pronounced at the time is not dramatically different to modern English. Yet Chaucer only lived two centuries before Shakespeare, and we live four centuries after Shakespeare. Why was there such a dramatic change in English at this time?
What Happened?
The difference in pronunciation that you are hearing is a result of what has been called "The Great Vowel Shift" (henceforward referred to as GVS). The GVS was a phenomenon that supposedly took place over the course of several centuries, starting sometime in the mid-14^th century and petering out in the 17^th century. The ending period of the GVS is generally considered the cutoff point between Middle English and Early Modern English.
Over the course of the ~300 years that it was occurring, the GVS changed the pronunciations of all of the long vowels in English. Originally the vowels adhered to "continental values". This is why Chaucer sounds vaguely Dutch. The vowels he was using still retained pronunciations established from the various continental settlers. Many of the continental languages have not undergone such drastic changes (at least within the same timeframe) and so we can still see similarities in pronunciation between them and Middle English. So what change did the English vowels undergo?
When comparing the continental and English vowel values, the main differences that you see are actually a function of how the vowel is physically formed by the mouth. If you know anything about linguistics you are likely aware of the concept of high, low, and mid vowels (along with back, front, and central). This is not an arbitrary designation. These terms refer to the position of the tongue within the mouth when articulating the vowel. During the GVS the long vowels moved higher (a process known, unsurprisingly, as "raising") with the highest vowels undergoing what is called "vowel breaking", which is a term for a monophthong turning into a diphthong. So without getting too bogged down in the specific linguistic factors going on, that's what happened. The value, the tongue height, of all of English's long vowels became raised.
This is not the only set of changes to occur, but it is arguably the largest. Over the intervening centuries between Shakespeare and now the main sort of shifts we have seen are mergers. To use the example from Wikipedia: meat and meet. Immediately after the GVS they would have been pronounced differently but are now homophonous (at least in standard English, but we'll get to that problem later). The GVS as a whole accounts for the difference between Chaucer and Shakespeare, with the ending years of the GVS and subsequent minor shifts and mergers accounting for the difference between Shakespeare and today.
How did it Happen?
I'm actually going to link you over to this answer by u/Muskwatch from a few years ago. Their analogy to music is a beautiful illustration of the mechanism at hand.
To summarize in a less beautiful way: There is a necessary distance between the pronunciations of the different vowels. Once the first vowel shift happened it set off a chain reaction, dragging or pushing all of the other vowels around to maintain the distances needed for proper differentiation.
Why did it Happen?
It is impossible to say with certainty. Languages are extremely complex entities, who are subject to the whims of an ever changing populace. It is hard to say why exactly a language underwent a change within the last hundred years (when we have more or less all the contextual information), and the GVS started 600-700 years ago. That is not to say that we cannot make some educated guesses about some of the precipitating events though (All info in this section comes from Jeremy Smith).
A major contributing factor appears to be the rise of London as a cultural hub within England. The GVS is thought to originate and radiate out from the South-East quadrant of England and during this phase London was solidifying as a center of government and trade. Now, in the more agrarian sections of the country it can be seen that residents had very close, very tight cultural ties. This creates an environment where each area has a particular, fairly rigid dialect just from the societal pressures. In an area like London, where there is a less tight-knit society and consequently lower societal pressures, it opens up language (and other cultural factors) to extensive change. It becomes more fluid from of an ever-changing cultural and social environment, with outside influences from foreign traders and internal influences from a rising middle-class where there used to be more or less just the agrarian and ruling class. But then because of the prestige associated with a hub like London, the changes begin to spread outwards as people looking to earn prestige adopt what they see as prestigious.
Another major factor in the South and Midlands has to do with some other linguistic changes. I'm not going to really get into the details here because I'm not a linguist so don't fully understand them myself, but many dialects in the south of England underwent a change called "Middle English Open Syllable Lengthening". This, combined with loss of inflectional endings left over from Old English, appears to have led to phonemicisation. This is where the choice between an allophonic pair changes the meaning of the resulting words (think the [s] in house for the noun and the [z] in house for the verb). This phonemicisation applied to the vowels a-, e-, o-, producing a different vowel value. This change can be seen in writings by Chaucer. But this change did not apply itself in all dialects and there appears to be a chase of sorts. Due to this phonemicisation (and a lingering influence from French) the more courtly classes had particular ways of pronouncing their vowels which the aspiring middle class tried to copy, prompting the courtly class to shift away again.
And this is all ignoring the changes going on in other parts of the country. It is all a big mess of interconnected factors dealing with normal linguistic operations and societal interactions all taking place over hundreds of miles and years. The idea of the GVS itself is problematic in that regard, trying to apply to a whole language and all its constituent dialects. I go over that a bit in the next section.
If I could join on a related question:
Are there studies on how much Shakespeare has anchored English in place? I remember Isaac Asimov asserting that English as a language doesn't want to drift too far from Elizabethan English precisely because that would make Shakespeare unintelligible. But was Asimov just saying that or have actual linguist historians made this argument?