For an interesting and amusing account of this see Bill Bryson's book 'Made in America'.
The modern US accent developed from the 1600s British dialect that was more like some 'regional' ones we may hear in Cornwall and the northeast of England today.
Interestingly, many of the US words like trash (rather than rubbish) and bug (rather than insect) and others are the ones that would have been used in the 1600s and have since fallen out of use in Britain.