I like to think of it as space contracting more and more as we draw closer and closer to the source M of the gravity field causing that contraction. So normal space would have the speed of light C = dS/dT; where dS is the change in location (distance in normal unbent space) and dT is the normal, undilated time that goes with normal space.
But as we draw nigh to a big M emitting a huge gravity field, space contracts from dS to ds < dS. But, this is important, to keep C up to speed no matter what, which is a fundament tenet of both theories of relativity, we must also have dt, which is less than dT > dt. That results because we must have ds/dt = C = dS/dT to keep the speed of light the same no matter what; so if ds < dS under extreme gravity, then C dt = ds < dS = C dT and dt must be < dT. Time must slow down relative to the outside as space contracts.
Naturally as dS ----> 0; dT -----> 0 as well to maintain the speed of light. So in extreme gravity, like in the middle of a black sphere (a 3D black hole), where space has been reduced to near Planck size, time all but stands still. This serves to explain why black holes are so long lived; time for them and relative to us outsiders watching them has stood still.
As space contracts more and more, its gradient gets steeper and steeper. This accounts for the gravitational effect we call acceleration. The steeper regions of space will invoke higher rates of acceleration on mass m that is caught in the gravity field of a mass M. But at no time, does the contraction turn inside out or fold in on itself as you suggest. If it did, time would also need to adjust to keep C at C no matter what.