Lagrange regions are where 3 forces cancel out: earth's gravity, moon's gravity and centrifugal force. Centrifugal force isn't actually a force, it's inertia in a rotation frame. It's the outward push you feel on a merry-go-round or a car taking a corner fast.
So when you're getting pushed outwards on a merry go round, it's not an outwards push (from the center), but a desire for the body in motion (you) to continue along it's movement vector, which is normal to the axis of rotation. The "centrifugal force" that our bodies feel is the cancellation of that force by a restraining harness.