I am still trying to wrap my mind around the run of black hole discussions I have seen recently. Fortunately, my cat Bella has been helping me out.
She likes playing with a shoelace. It doesn’t matter where I put it; she will find it and drag it out again. If I throw it in the trash, she will be on it before the echo dies down, showing only her hindquarters and tail as she digs it out. After she has lost interest in the shoelace, it is no longer straight.
Saying that the shoelace is not straight though does not really tell you much about it. She may have left it wadded up in a corner, draped over the armchair with one end dangling, or just left as a double strand wrapped around the table leg because she likes to carry it from the middle.
In the same way, saying that a concentration of mass warps or bends space does not tell you much about the gravity. It does not tell you the shape of space after gravity has bent it. My own take on gravity is that it thins space in the concentric shells around a massive object. Any given shell will be thinner the next moment from now. Objects in that shell will then be closer to the massive object because the intervening space has decreased. The objects do not move through space so much as being carried along with contracting space, much as galaxies do not move through space as expanding space pushes the universe further apart.