When Earth was forming it was hit by another small planet, and some material splashed into space from that impact ended up going the right speed to orbit Earth as rings. That material eventually clumped together to become our Moon. Why didn't this material stay as rings

This is a consequence of what is know as "Roche Limit." The Roche Limit refers to a determinate distance from a planet, based on the mass of this body. Anything within this distance that is held together only by gravitational forces will be broken apart by tidal forces of our planet. This is the case with the rings of Saturn. As objects get closer to the planet, the tidal force difference increases.