Short answer: The Great Vowel Shift. This was a vowel shift which took place in English between around 1350-1700.

Let's start with a. So in Chaucer's day, the word name would have been pronounced something like 'naa-muh' (I assume you're not familiar with the IPA), as it is today in German. Over time, the middle vowel would have been 'raised' to sound more like 'nehm', and then diphthongised to the present pronunciation 'naym'. So that's how we get from 'ah' to 'ay'.

Similarly, east would have been pronounced something like 'ehst', but the vowel became raised to sound like 'eest'. Hence we go from 'eh' to 'ee'.

So what happened to sounds that at the time already contained the 'ee' sound, such as 'time', pronounced 'tee-muh'? Well, first the vowel became diphthongised to sound more like 'tuh-eem', and then the vowel was lowered to sound more like the present 'tah-eem'. Hence how we get 'eye' from 'ee'.

You get the picture for the other vowels.