There are many reasons: languages change and generally become simplified over time, trends emerge in pronunciation changes, etc. Often there is not one clear defined event that someone can point to and say: “This is why this happened.”
However, for English at least, there is one event that significantly changed English pronunciation. This is called the Great Vowel Shift. I will be using the International Phonetic Alphabet to show pronunciation.
Around 1350/1400, words were pronounced roughly the way they were spelt. The Great Vowel Shift changed this over a few hundred years starting around this time. In the first phase, the closed vowels /iː uː/ became dipthongs (where two vowels are pronounced by sliding to one another, rather than pronouncing them separately), turning into /ei ou/ or /əi əu/, respectively. At the same time, the mid-close vowels (not as close as a closed) /eː oː/ underwent a raising process (meaning that the placement of the tongue is higher in the mouth than it was before) to become the new, closed vowels /iː uː/.
This resulted in the following:
The second shift saw more open vowels (like /a: ɛː ɔː/) become more closed (/eː iː oː/ respectively). However, this was not a direct path from A to B. It was more like a winding path from A to G to Z and then finally to B. Over the course of a few generations, great-grandparents would not have been able to understand their great-grandchildren as easily as we could have communicated with ours due to the vowel shifts.
In the second phase, we see the following pattern:
So what does this have to do with homophones (words that sound the same)? As all of this shifting was occurring, vowel sounds were actually shifting to sound the same. Before this phase, meet and meat sounded differently. Afterwards they had become homophones.
Wikipedia has a great set of audio files on how the following words were pronounced over time. The first pronunciation is from 1400, the second from 1500, the third from 1600 and the last from 1900. The modern pronunciation is based on what is considered to be a “standard” pronunciation.
Naturally, because this is the English language (although the same would have occurred in most languages), there are exceptions to this shift, such as father, which still is pronounced the same way as it was in Middle English.
Here is some more information about the Great Vowel Shift if you would like to go into even more detail.