It should be noted that the American and British accents of today came from a common ancestor. This means that contemporary British is as different from 18th century English as contemporary American English is.

The "standard" you are asking about is a very interesting question. Part of the revolutionary movement in the United States was to establish it's own English standard. Webster's first dictionary was an integral part of these reforms. A product of this movement is the spelling reforms which removed the 'u' from many words. Colour became color, labour became labor, armour became armor, etc. The switch from '-re' to '-er' endings in words like "center" and "theater" were also part of this movement.

What many consider the "standard" American accent today is basically a mid-western accent, but even here we run into trouble as there are still regional accents like the Chicago accent (da Bears!) or the Minnesota accent (ya dontcha know!?). There is a theory that, for the most part, the standard accent followed migration patterns west from New England, into Ohio, across the Mississippi, and into California and the Pacific Northwest. The more mobile a population is, the less chance there is for a regional accent to develop. Which is why there are so many more regional accents in the tiny British Isles then there are in the much larger area of the US and Canada. British people tended to stay put much more than the English speaking populations of the US and Canada.

Another factor that should be mentioned is the rise of mass media in the mid-20th century. For whatever reason, national broadcasters tended to have the mid-western accent, which may have led to the concept of it being the "standard." Another idea is that the majority of US Presidents came from areas where the mid-western accent either originated or migrated to. I believe more presidents were born in Ohio than in any other state.

Sorry, this was kind of off the cuff. I'm just going off of a casual interest in the subject. Hopefully a linguist can chime in.