Look a little closer
Soap removes grease not by making the oil vanish, but by helping water carry it away. A soap molecule can do this because one part interacts well with water while another part is attracted to oily substances.
Plain water struggles with grease because water and oil have different patterns of molecular attraction. Water molecules prefer one another, while oil gathers with oil. Even after rinsing, a thin greasy film can spread across the surface again.
The oil-friendly tails of soap molecules push into the grease, leaving their water-friendly heads facing outward. When enough molecules gather, they can surround a tiny droplet of oil in a roughly spherical structure called a micelle.
The outside of a micelle interacts with water, allowing the trapped droplet to remain dispersed instead of immediately joining the larger grease layer. Rubbing breaks contamination into smaller patches and gives soap access to more of the surface. Rinsing then carries away the micelles and the oil held inside them.
Not every stain is removed by this exact mechanism. Protein, mineral scale and some pigments may require enzymes, acids or other cleaning chemistry. Hard water can also reduce ordinary soap's performance when calcium and magnesium ions form an insoluble residue with it.
Soap is therefore a molecular go-between. By gripping oil at one end and interacting with water at the other, it lets two normally reluctant partners take the same trip down the drain.
