Milk is mostly water, but it has some milk fat. Fat and water do not mix very well (fat is not soluble in water), so the molecules of fat in water form tiny balls that are suspended in water. This situation is referred to as a colloidal suspension and it is rather common. When the food coloring is dropped into the milk the tiny balls of fat in the colloidal suspension prevent it from spreading throughout the container and they stay, more or less, in one place.
The detergent changes everything because it is capable of mixing (dissolving) with BOTH the fat molecules and the water molecules. As the detergent dissolves some fat molecule balls others move in to replace those interacting with the detergent. They carry the food color molecules with them and the whole system seems to churn with amazing colors. Only one drop of detergent is needed. If the churning stops, there's a good chance the detergent has been "used up". It has interacted with as many fat molecules as it can. Adding another drop of detergent starts it up again. You can design an experiment by trying to relate how long different size drops of detergent keep the churning going!