Good call. Looks as though you're right.
Why do we need the "cio che" isn't that "that what" or "what what" ?
In this context, the closest equivalent is 'that which' - 'you don't understand that which I want to say' (or, less literally, 'you don't understand the thing that I mean'); in English, we frequently omit the 'that' and just say 'what', which is how the answer is translated. There's an interesting discussion at http://forum.wordreference.com/showthread.php?t=226708. I've also heard somewhere that 'quello che' is much more common in speech, while 'cio che' is more common in written Italian (generally considered essentially interchangeable).