What is the direction of the processing of light by the (human) retina and how does it happen?
Short answer

Photoreceptors are the only cells in the retina that directly detect light; they can detect light through the other layers because those layers are so thin and effectively transparent: light passes right through to the photoreceptors.

Longer answer

The retina is fairly thin, and the bipolar/RGC layers don't contain much pigment. Have you ever looked at a thin slice of tissue under a microscope? You can do this and focus to different layers because single layers of cells are fairly translucent (remember they contain a lot of water): most light passes right through them. Some cells, such as red blood cells or photosynthetic plant cells, are harder to see through, but that's only because they are full of pigments.

The reason the photoreceptors respond to light in a way that can be detected biologically is that they contain special photosensitive pigments. In rods, this pigment is called rhodopsin and it is composed of a protein and a molecule of retinal. Most other cells in the retina don't have any such photoreceptors (there is a special class of retinal ganglion cells that is an exception, see for example here - these cells have a pigment melanopsin and contribute to light detection for circadian rhythms rather than vision).

Other cells in the retina respond to light only because they are sensitive to neurotransmitters released by photoreceptor cells and other retinal cells.

As far as why this is the arrangement, with photoreceptors in the back, it's really just a fluke of evolution and evidence that evolution and not design is the best explanation for diversity in biology. Eyes have evolved more than once in animals, however, and cephalopods like squid do not have their retinas arranged this way; in squid the photoreceptors are the most superficial layer.

Because the retina is so thin, this arrangement doesn't provide a large enough disadvantage to be selected against given the complexity in changing the organization. One exception to the thinness, however, is in the blind spot. The blind spot exists because all of the axons from the retinal ganglion cells have to exit the retina someplace. They exit in a big bundle, and there can't be any photoreceptors where that bundle is, so there is a spot where you cannot see. However, the position of the spot is not in the center of your vision, and that part of the visual field is usually covered by the other eye