At some point I was translating online between English and German (which are very similar languages linguistically), and I noticed the following curious phenomenon.
English - German
street - strassen
hot - heiss
nut - nuss
Each of these words, and many others, is essentially the same in English and German. The only difference is that the "t" sound in English is replaced with a "ss" sound in German.
This is, of course, the exact difference between the "normal" and Ashkenazi pronunciations of Hebrew!
My theory is that, about a thousand years ago, everyone in Germany forgot how to say the letter "t" and started pronouncing it as "s" instead. So words in German that had a "t" in them were now pronounced differently, and eventually this was recognized through the use of a new "ss" letter. For the Jews in Germany at the time, this affected their pronunciation of Hebrew as well as German. Based on the common German pronunciation, the letter "taf" became a "saf".
These Jews and their descendants moved east and became the Ashkenazi population of Europe. Despite their migration and dispersion, they remained remarkably faithful to their customs of speech. Their day-to-day language remained a variation of German (Yiddish), and their pronunciation of Hebrew retained the oddities of the non-Jewish population their ancestors had lived among in Germany.
Today Yiddish has virtually died out, but if you want proof that Ashkenazim indeed have ancestry in Ashkenaz, look no further than the "Ashkenaziss" pronunciation which remains alive and well to this day.