The actual material the paper is made out of (mostly cellulose) is transparent. Because it’s denser than air, light bends when it passes through the boundary between air and cellulose (index of refraction). The surface of the fibers is very rough, which means that light passing through a fiber gets scrambled, as any slight difference in angle or location translates to the light exiting in a totally different location.
Water fills in the gaps, and is pretty close to the density of cellulose. It makes something much closer to a flat surface like a pane of glass, which means light all gets bent in the same way, and produces something much closer to a recognizable image after it passes through the paper.
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