What activates them is public opinion (I know this because I've coded extensively).
When public opinion (average across all views) reaches 60%, the CCS will enter the state CCS_APPEARS, wherein the CCS will appear but won't conduct terrorist operations. You can still encounter them, though, and they can raid you, just not the rest of the world. They can only attack you if your safehouse either (1) has heat, or (2) has both compound walls and a printing press, however; low-profile safehouses are immune.
At 80%, this upgrades to CCS_ATTACKS, whereby the CCS will start conducting terrorist attacks. They also become better equipped, and more common in the world.
At 90%, this upgrades to CCS_SIEGES, at which point any LCS safehouse becomes at risk from a CCS siege, even safehouses with no heat, no walls, and no printing press. They will also become even more heavily armed and common.
Killing a CCS base leader and capturing the boss' safehouse will knock the CCS back one step, but they will recover after one month. If you somehow manage to take public opinion from less than 60% to more than 90% in less than a month, it will still take three months for the CCS to reach the CCS_SIEGES stage, as it can only move up one step at a time, and only at the beginning of a new month.
There is effectively no way to win the game without waking up the CCS short of disabling the CCS at the beginning of the game (this sets the CCS state to CCS_DEFEATED, which is the same state you get if either you or the FBI captures the last CCS safehouse). The List of CCS Backers cannot be acquired while the CCS status is still inactive.