A better example of phrases that mean different things on either side of the Atlantic is what it means to "table" a proposal. In the US it means "kill it", but in the UK it means "bring it up". If an American says "could care less", they're just being incorrect. ;)
"could care less" is a shortening of "could not care less", and the latter phrase is correct. the words themselves don't mean anything any more, it's a stock phrase.
i think the sentiment is, "I care so little that I don't even want to use all the words needed to say how little I care".
http://en.wikipedia.org/wiki/Table_%28verb%29