Why do Americans find the British accent rewarding to listen to?

Because we over-enunciate everything! Same language, just sounds slightly snobbier!

An alternative perspective from a(nother) Brit:
Except of course for those of us with regional accents who drop our "aitches" and sound not in the slightest bit posh or snobby! However I think that's the point, the small but important differences that make varying accents interesting and appealing.

See the links for a bit of a compare and contrast exercise (links to Youtube videos):

1. Hugh Laurie (Eton and Cambridge educated posh accent) and Ellen Degeneres.
2 & 3. Fred Dibnah (Lancashire Bolton accent - chosen primarily because it shows the diversity of British accents especially when compared to Hugh Laurie) climbing an old mill chimney and felling one.

People obsessed with health and safety at work and personnel protective equipment may find they want to hold their head in their hands and rock slowly backwards and forwards while watching the Fred Dibnah videos as there are no hard hats in evidence, no high vis, no safety harness / rope, no secure cordon and no siren to warn of the impending chimney collapse. He does however have a whistle, a flat cap and a fence post to lean on less than 30 feet from the base of the chimney...