Washing with soap and water is a natural and common way to achieve cleanliness.
Water has a high surface tension.
Soap helps to lower the surface tension of the water by interacting with the water and forming a mixture called an emulsion. Scrubbing with soap helps to perpetuate the development of this emulsion and create negative charges which surround the dirt particles trapped in the soapy mixture. The mixture of soap and water through the action of scrubbing keeps the dirt particles in suspension. Running water can then rinse the whole mixture away leaving the hands clean and refreshed. The best news is that textbook knowledge translates to real life benefits.