A soap has two dissimilar ends. At one end it has the hydrocarbon chain that is non-polar and hydrophobic (soluble in oil/dirt). At the other end there is the carboxylate ion that is polar and hydrophilic (water soluble).
When soap is added to water, its molecules make a unimolecular film on the surface of water with their carboxyl groups dissolved in water and the hydrocarbon chains standing on end to form a hydrocarbon layer as shown in the figure below.
When a dirty cloth is soaked into a soap solution, soap dissolves fat or oil with dust by micelle formation. The fat or oil with dust is dissolved in water by hydrophobic hydrocarbon chains of soap. The water soluble carboxylate ions make a hydrophilic surface in water and render the micelles of oil or fat from the cloth to the water. Thus, the micelle is dissolved in water and is washed away. Soap tends to concentrate on the solution surface and therefore lowers its surface tension, causing foaming. This helps it to penetrate the fabric. It emulsifies fat and dirt to form micelles and make all the micelles water-soluble. Thus, the water washes the dirt away.
