What does "CC (Charlie Charlie)" mean when pilots reply to ATC?
Short answer

This wording comes from the common meaning of letter C to be Correct/Yes, early standardized and used at sea since 1857. Under the Commercial Code of Signals (1857-1902), the C flag, in addition of representing the letter itself, had the meaning of Yes/Affirmative. This code has been improved and extended into the International Code of Signals (ICS). While the visual representation changed, the meaning was preserved.


Today's version of the C flag (previous version)

From the 2003 ICS:


  Signals for flags, radiotelephony, and radiotelegraphy transmissions.
  C: Affirmative—YES or “The significance of the previous group should
  be read in the affirmative”.


The pilots you heard were saying affirmative.



Longer answer

Codes have their origin in the need to abbreviate everything at a time the communication channels were limited, as for nautical flags, Chappe, Morse, Aldis, etc. Nineteenth and early twentieth centuries saw appearance of codes (flags, lights, telegraphy, Q, X and Z) in order to federate many isolated practices, in particular in commercial telegraphy (Marconi) and railroads. Codes have many resemblances as they have borrowed good practices and common usages from each others.

In radio communications, letter C has been early used in consistence with the ICS, to mean correct, like when acknowledging a readback with a single C, meaning this is a correct readback.



Example

Discovered by @ymb1, this accident report includes a transcription of the pilot (CA) controller (MA) exchange:


  
  CA: Roger, we will appreciate it if you can alert, er, 
  fire, er, er, er.  
  MA: Do you request a full emergency please? A full emergency?  
  CA: Affirmative, that's Charlie Charlie  
  MA: Roger, I declare a full emergency.  
  


later:


  
  MA: Confirm runway one four?  
  CA: Charlie Charlie.  
  


And this other document, still from @ymb1:


  
  Do you have ATIS information Delta?  
  Charlie Charlie we have Delta.  
  


...


  
  Ok our stand 41 AF028.  
  Understand B 41.  
  Charlie, charlie.  
  


Details

Charlie-Charlie is a fancy substitution for a standard affirmative and accepted non-standard words roger and wilco.

Standard phraseology is meant to prevent confusion, and has been improved after incidents/accidents where the communication between parties played a role. In a conversation listed above, we can imagine how information can be  easily misunderstood:


  
  Do you have ATIS information Delta?
  Charlie Charlie we have Delta.
  


ATCO are not required to know the ICS, though in many countries, an important part of civil aviation ATCO have had a military career, and are likely to know the meaning of letter C in many codes.

More


More on signals, and how to use flags: ICS, US edition, revised 2003.
More on codes and abbreviations: Origin Of Ham Speak -  Fact, Legends And Myths (AC6V)