As it was mentioned in Laws: Royal laws, an investiture is a law which affects directly on the possibility of nominating bishops and their taxes. If you have a free investiture you can decide personally about residents of bishoprics and collect taxes from them (if bishops agree). When the investiture belongs to the pope or patriarch, all decisions are theirs. It's not hard to guess that the first option is more cozy, but it dissatisfy the Head of the Church. It hasn't end with an excommunication or an open war: usually the religion leader limits to ask for the change of the law. Only in the moment of refusal, the difficulties begin. That's why it's worth to change some royal law just after getting the crown: after that you won't be able to change another one (but your descendant can).
If you have a free investiture you can nominate new bishops in the bookmark of religion in the upper part of the screen (screen above). If one of your vassals is the servant of the Church and is making troubles, it's worth to choose good successor and kill the troublesome subordinate. No civil wars guaranteed.
Matter of income from the Church lands is explained in the chapters Economy and Laws: Royal laws.