The evolution of the North American accent is complex. The different regions of the original colonies were settled by Britons who already had regional British accents: the puritans from East Anglia who settled New England, and the settlers from the West Country (Cornwall, Devon, Dorset and Somerset) that settled the coastal South. These were soon influenced by the Dutch around New York and New Jersey, and by African dialects in the South.
Because these regional pockets were relatively isolated from each other over the many decades when transportation and voice communications were minimal, they held onto their distinctive characters and also gradually drifted away from their British phonological roots. Vowel sounds slowly shifted, sometimes even losing distinctions among vowel sounds that their British forebears would have made. Over time, large urban centers developed quite distinct accents, many of which have survived in muted form today.
The greatest initial factor in the mixing of accents was the push of Western settlement. As the bearers of the regional Eastern and Southern accents diffused westward, the accents became less distinct. There was indeed an “averaging out” of accents in the Midlands (though the Scandinavian influence in the Upper Midwest introduced a marked regional character).
I should note that all of the above is a rather gross simplification.
The second greatest factor in the blending of accents, of course, is the high mobility of the modern population and the ubiquity of voice communication across regional boundaries.