From a newtonian perspective you can imagine that gravity emits a force between all pairs of massive objects. That's a good way to look at it for small values of mass. It works pretty well on the scale of our whole solar system. This is what you're imagining when you say "not by gravity but by spacetime curvature".
But gravity is spacetime curvature. It is manifested by changes to spacetime itself made by that mass. For very large values of mass (or density in this case) that newtonian approximation doesn't hold according to experiment. We need a different model and the model that we've found that works is that mass changes the shape of spacetime. Its presence changes the nature of straight lines and the paths that you can follow (geodesics).
A black hole causes all possible paths of spacetime, paths that light must follow because it travels through spacetime, to point into the black hole https://en.wikipedia.org/wiki/Black_hole#Event_horizon Light must follow straight lines but because the space containing those lines is curved, there don't exist any of those lines that point outside of the black hole. Once you've crossed the event horizon, space itself doesn't have any straight lines that point outward, there are no paths that you can follow.
It's even weirder than this because we're mostly familiar with motion through space so that's probably what you (and I) are imagining, but this is motion through spacetime. So really the correct thing to say is that there exist no 4-d paths that point outside of the black hole that are also in the future. And that's even weirder than it sounds because you asked about light specifically and light doesn't experience time the same way you and I do. So our intuitions and our ability to map reality to the nice charts that we're used to looking at are pretty limited here.
