Mixing food coloring, dish soap and milk causes the molecules in the milk to separate. The food coloring then mixes with the milk, creating a colorful result.
The dish soap is essential to cause the separation reaction to occur. When mixed with milk, it bonds to the fats it contains, creating micelles that cause the surrounding molecules to be pushed away. This lowers the surface tension of the milk and allows the food coloring to spread throughout the liquid.
The fats in the milk are important to cause a stronger reaction. If the milk has a lowered fat content, it results in a weaker, less visually pleasing result.