The idea is that you put someone into a fake reality by controlling all the information they get access to.

It's actually very hard to do to an unwilling target, because all it takes is one bit of real reality slipping through the cracks to bring the whole thing crashing down... However, it can be done quite well if you condition the target to be willing first, like we see with the Left. It doesn't matter when reality knocks on the walls of the illusion, because they prefer the illusion.