A black hole is not a cosmic drain. It is a place where the geometry gets so steep that light runs out of legal directions. The cleanest way to see that is to follow the light itself, right up to the moment it has no route home.
Light travels in straight lines through the space it is given. Mass bends that space. Far from a star the bend is a tiny correction; near a black hole the correction becomes the whole story. A ray can glance away, whip around in a hairpin, loop near the photon sphere, or disappear through the horizon.
Here is a single ray on its own. The black disk is the event horizon, the point of no return. Aim the ray closer or farther from the center and watch how hard it bends. Orbit around to see the path curve through 3D space.
At one exact distance, one and a half times the radius of the horizon, gravity bends light by just the right amount for it to travel in a circle. A ray arriving there, aimed perfectly, will orbit the black hole.
It is a desperately unstable place. Nudge a photon a hair inward and it spirals down through the horizon. Nudge it a hair outward and it slowly winds away to freedom. But light grazing this radius can loop the hole once, twice, or many times before leaving, which is why the bright ring in the first figure is really the same light wrapped around and stacked on itself.
Crossing the horizon is not like hitting a wall. Locally, nothing special happens there. What changes is the geometry of the future itself.
Every event has a set of directions light can go from it, its light cone, and nothing can travel outside its own cone. Far from the black hole the cone points evenly toward the future, and you are free to move outward or inward. As you get closer, gravity tips the cone over, toward the hole. At the horizon it has tipped so far that even an outgoing ray of light, the fastest possible escape, only manages to hover in place. One step further in and the entire cone points inward. Every possible future, for light and for you, leads to the center. There is no direction left that points out.
Go back to the first figure and fly around. Because the hole bends every ray that passes near it, the disk of glowing gas does not sit flat like Saturn's rings. You see its far side lifted up over the top of the hole and its underside curled beneath, because the light from those regions is bent around to reach your eye. The dark circle in the middle is not the horizon itself but its shadow, swollen by all that bending to look markedly larger than the horizon really is.
None of this needs exotic math to picture. It is one idea: mass curves space, light follows the curve, and a black hole curves it so hard that some paths only point one way.