Can someone please explain how "I could care less" and "I couldn't care less" can possibly mean the same thing?

As others have said, they don't mean the same, but usage in the US in particular has overcome logic. The same with 'could have' or 'would have' being corrupted to the nonsensical 'could of'/would of'. This happened because of mishearing 'would've' and 'could've' and the users of such illiterate expressions not being corrected.