How did it come about that Americans pronounce words with the letter "i" as "eye"? As in "anti" and "Iraq",?

Americans pronounce "i" two ways - the long way "eye" and the short way "ih".

when the "I" is at the beginning or the end of a word (unless it's the only vowel in the word, such as "it") it's pronounced the long way. When it's in the middle of the word, it's pronounced the short way.

Our pronouncing it "eye" rather than "ee" like the rest of the world probably comes from the Latin and Greek contributions to English (phillipi, octopi, Ides of March).

It's just their way of thinking that it's correct. They don't want Anti to be pronounced like "aunty/auntie", which would cause confusion. There's no such word as "i-rack", or "i-rahk". If Americans were able to pronounce the q like Iraqis, then they won't be pronouncing it like it is. It's just their way of saying "I'm unique". This is mostly from those with either:

No education, very little education or from military speak.

Ok I see that you are Canadian and Canadians also do that and did that way before the ipod was invented.

You should stop having so much hatrid for a country that is almost identical to yours.