You aren't missing anything. "I could care less" is quite simply incorrect usage in almost every instance. People say it when they mean to state that they do not care at all about some issue. But, as you rightly point out, the actual meaning of the phrase is quite the opposite of the speakers' intent.
I'm a native speaker of American English, but I've also been baffled as to why so many Americans use the phrase. I guess it is just a set phrase, and people don't think about the true meaning of the words they are speaking. How it got that way, I don't have any idea. Maybe we are in too much of hurry to waste our time enunciating the extra syllable.
I tried on a few occasions to point out the error of this statement, but I generally received blank stares in return. Either that or an increasingly hostile exchange that went something like this:
speaker: "I could care less!"
me: "Oh, then you are concerned about this matter?"
speaker: "No, I just said I could care less."
me: "Exactly - so you must care about it to some degree."
speaker: "Why aren't you listening to me? How many times do I have to tell you that I could care less?!?"
It is quite pointless to fight it, as it just annoys the speaker. Besides, the intended meaning of the phrase is well understood. It's common usage, even if it is nonsensical usage.
cfitz