Wife asked the question last night. We theorized, but I thought someone here could provide a proper explanation.
There is always room for discussion, but perhaps the American and British accents section of our FAQ will answer your inquiery.
I'm not super knowledgable on this subject but I do have some basic knowledge of it. In some ways, the question is wrongly phrased because from what we know about how the English accent sounded in the 16-18th century, you could almost ask the question in the following way: "Why did the English stop sounding like Americans?" This is an oversimplification, but let me explain.
Traditionally, the English accent was what linguists call a "rhotic accent". This means that average English men and women in the 17th century pronounced their "Rs" with a hard "r" sound, like Americans do, as opposed to the soft R spoken by British people today. When the British began to colonize the Americas, they brought this accent to these shores. However, as we move into the 17th and 18th centuries, the British upper class began adopting a distinct dialect which employed a non-rhotic accent (meaning, they dropped the hard Rs) to distinguish themselves from the lower classes. Over time, this caught on with the rest of the populace (this same thing happened with the "thou"/"you" distinction. We often think of "thou" as very formal, but it was actually considered INFORMAL English. But, in the end, upper class ways of speaking filtered down to the lower classes over the centuries until familiar forms like "you" became more common, despite being considered previously formal language). But, while all this was happening in Britain, colonial English was evolving independently of the changes that were occurring in Britain. That is how the differences came to exist: Americans kept the rhotic pronunciation, which is why the British later mocked Americans for sounding backward and old-fashioned, while the British moved on to a non-rhotic accent.
Here are some links to hear what i'm talking about:
https://www.youtube.com/watch?v=gPlpphT7n9s (Shakespeare spoken in 17th century variety of English, the rhotic kind).
https://www.youtube.com/watch?v=AIZgw09CG9E (Tangier Island, VA, an island in the Chesapeake, whose people have lived in isolation and preserved many of the same mannerisms of 17th century varieties of English).