Why do the legs of a person in the water appear to be shorter to a person who stands on the bank of a river?

Because light, at the interface between two media – water and air, is refracted.

Moreover, when light passes from a less dense medium to a denser one, the angle of refraction becomes less than the angle of incidence.

Accordingly, when reflected, light from a smaller angle of refraction is projected into a larger angle of incidence. The result is a distortion of the light image.
That is why it seems to us that the legs in the water are shorter than on the river bank.