The key to monuments is to build them out of a color that you don't care about, and use them to produce a color that you do care about. If I have many more black VPs than blue VPs, I'll try to build a monument out of black tiles in a kingdom in which I have a blue leader with solid support. Giving up the black support risks me nothing (in fact, it removes the possibility that I'll lose a black conflict in that kingdom to someone who really needs black points and give him 4 black VPs, because those supporting tiles aren't there anymore), and if I have a good enough base of blue support I can harvest blue VPs for quite a while.

The way to create this situation is through leader conflict. (After all, if I'd been drawing enough blue tiles to build a kingdom with a well-supported blue leader, I wouldn't have a shortage of blue VPs.) You need to look for a kingdom that has supporting tiles in a color you need and the possibility of creating a monument in a different color. Then you have to drop your leader in, win the fight, and build the monument to start harvesting the VPs you need.

The remarkable thing about E&T is that while the situation I just described sounds pretty abstruse, it actually happens pretty often. But a lot can go wrong.

Another thing to bear in mind about monuments is that they're catastrophe tile magnets. It's often much easier to wall a leader off from a monument than it is to pick a fight with him. So watch out for situations where your leader can easily be cut off from the monument you're harvesting, and avoid getting into that position in the first place if you can.