Why does soap remove oil?
The short answer
Soap molecules have a water-loving end and an oil-loving end. They surround grease droplets so water can carry them away.
How it works
Oil does not mix with water by itself. Soap lines up at the boundary, disperses the oil, and washing and rinsing remove it from the surface.
A familiar example
Greasy dirt that resists plain water may be easier to remove with soap and warm water.