Firstly, the original English-speaking inhabitants of America would have spoken various different dialects of English, and most would not have spoken standard British English, as that was the speech of the wealthy classes who don't often emigrate. So already there was a very different speech community from that of Britain.
Secondly, sound changes happen within all speech communities, and they are to all intents and purposes random. The vast majority of British and American English speakers didn't ever communicate with each other, so the languages developed in different directions.
As for the specific differences, it's probably not possible to say where most came from (random changes, see) but we can guess at some. For example, standard American pronounces the 'r' in 'air', while standard British doesn't. But many British dialects do pronounce the 'r', so we can assume that speakers of at least one of those dialects were represented in significant numbers among the early immigrants to America.