The participants create fresh addresses A', B', C', D' but do not show them to each other. The goal of CoinJoin-based mixing is to create a mixing transaction with input addresses A, B, C, D and output addresses A', B', C', D' to hide the relation between the coins and their owners. (If it is not clear to you why such transactions are possible, I recommend reading the thread about CoinJoin). However, if we would stick to that particular order A', B', C', D' of output addresses, everybody would learn that A belongs to A', B belongs to B', and so on. So we need to shuffle the list of output addresses to make sure that the linkage of input and output addresses remains hidden. But just shuffling the output addresses in the created transaction does not suffice: For example, if everybody just announced his output addresses during the protocol in plain, i.e., Alice announces A', everybody would learn that A' belongs to Alice. So we have to make sure that the messages that are sent during the protocol do not break the anonymity. CoinShuffle solves exactly this problem.