From my understanding many soaps contain sulfactants and surfactants that disrupt the sulfur and surface tensions in the outer membrane of bacteria cells.

This disruption, in addition to lipid interactions and mechanical abrasion, makes many bacteria weaker; when the bacteria is then transferred plate to plate and to sponge it is less able to survive or reproduce.

Also after bacteria interact with soaps, regular water can act as a mild solvent further degrading the bacteria (and washing it away).