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That shader is actually fairly simple, it takes a "number of colors" parameter, then multiplies the color on screen  by that number, rounds the result (removes decimals), and then divides the result by the number of colors.

Since in Godot shaders each color (red, green, blue) is represented as a value from 0 to 1, the shader sort of "flattens" the colors in similar range.

The weird "moving/pulsating dither" effect is done by messing with the number of colors in-game.