With gravity we notice that everything is affected in a way independent of the mass. A light object and a massive object with the same velocity move the same under gravity. And the motion is one to converge lines together (if everything is drawn towards the center of the earth, they are drawn together). A curved spacetime does exactly that, just like all paths from the north pole converge on the south pole, if we have a curved spacetime then everything can converge due to curvature, regardless of whether it is light or heavy.
Once everything moves that way regardless of mass, then light itself is forced to so itself. The spacetime outside a black hole is curved, so that affects the light outside a black hole. The curvature outside the black hole was fixed by the matter than collapsed in to form the black hole, and it set up curvature that begets equal curvature, that persists in time. For an event horizon, those paths (that work just as well regardless of the mass of the thing following them) don't get any farther away from the black hole, so nothing (not even light) can escape because everything is affected in away independent of how much mas it has.