Here's an intuitive explanation for L1:
Imagine you have two planetary bodies fixed in space. A free mass is attracted to both, but it's possible to find an unstable equilibrium somewhere between them. Put the mass here, and it'll stay still-- but it's like balancing a ball on the top of a sphere; it'll roll off towards one of the two planets unless you stabilize it.
Now imagine displacing the mass from the equilibrium, but in a direction perpendicular to the line between the planets-- it'll swing back and forth like one of those slingshot rides. There are two dimensions perpendicular to the line between the planets, so two dimensions with stable oscillations, therefore you can get circular orbits.