what keeps a rocking chair from tipping backwards?

Answer 1

The rockers center of curvature is designed to be higher than the center of gravity of the chair and of the chair + rider combined. So, rocking the chair raises the center of gravity, and an increasing amount of force is needed to increase the rocking angle. The very tip of the rockers can even have a decreasing curvature, increasing the the rate at which the C.G. moves up, which lessens the possibility of tipping over.
Of course, this auto-stabilizing property does have limits that can be overcome if someone really pushes it...

Answer 2

the rockers are long enough, and so shaped that the center of gravity of the chair and rider cannot get back past the tip of the rockers.

i IS of course possible to tip a rocking chair by applying unusual force.