FYI, you have it backwards--the standard or typical American accent is rhotic, with some exceptions (Boston, AAVE/Ebonics). The non-rhotic accent was in the very slow process of becoming standard in England as America was colonized, so really, it's not a question of America "becoming" rhotic but rather why the movement continued in Britain but aborted in most of America.

Boston/New England and some southern cities that maintained close ties to England in the late 18th/19th century were more influenced by speech patterns and still drop the r in many cases. AAVE's non-rhoticity is generally considered an artifact of the non-rhoticity of pre-WW2 Southern accents. Levine and Crockett have argued that the postwar shift to pronouncing the /r/ in the South reflects a change in intellectual prestige: rhoticity became viewed as educated and elite. (Erik Thomas has posited that part of this shift was motivated by a desire among white southerners in the civil rights age to continue marking themselves as superior.)

As typical, the first signs of change are visible in the speech and writing of women, especially young women.

This question might be better suited for /r/linguistics