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Refraction is just the best.
When light goes from one material to another it refracts, or bends.
But refraction alone doesn't explain why the arrow reverses. The glass of water is also acting like a magnifying glass.
Physics Central adds: “When light goes through a magnifying glass the light bends toward the center. Where the light all comes together is called the focal point, but beyond the focal point the image appears to reverse because the light rays that were bent pass each other and the light that was on the right side is now on the left and the left on the right, which makes the arrow appear to be reversed.”