
Ben M. answered 04/12/19
Japanese/ESL tutoring from an accomplished translator/interpreter
This is a good question!
In a nutshell, the American accent is most closely related to the Irish accent. When the US colonies were first being formed, people from different parts of Britain ended up in small communities in places like modern-day Virginia, etc. They all had different accents, but the adults didn't change they way they spoke.
Their children, however, did! There's this interesting Linguistic process called "leveling," which occurs when you have many children from different linguistic backgrounds all trying to speak the same language. In the case of the same language having many different accents, children will "level" the accent, meaning the kids will adopt a uniform accent amongst themselves, even if it differs from the accents their parents use at home. If you think about this from the parents' perspective, it's really interesting and bizarre: Your kid speaks the same language as you, but talks in a markedly different way.
So, basically, one of the biggest influences over these early days of the American accent was Irish English (I can't speak specifically as to why this was, as I have more background in Linguistics than History). When the colony kids were "leveling" what eventually became American English, one of the characteristics they left in was pronouncing [r] from Irish English. Hence, American English is rhotic.
Just to round out this answer, in addition to Irish and American English, Scottish English also pronounces [r].
I hope this helps!