There are two things to note here. The visible foam you see is a product of the chemical structure of soap (surfactant) that others have described. The important “foam” is microscopic, and these “bubbles” called micelles, like u/yalloc describes in their comment. The reason this is so important is that oily dirt does not dissolve in water that well. But these micelles encircle the dirt and bond to it very well, and the outer side dissolves in water very well, making it easier to rinse it away. These microscopic micelles do tend to group together making some visible foam, but not all foam is made up of micelles. Regular old bubbles from foaming agents do make it easier for the liquid to cling to the washing surface a little better, and seep into it, instead of dripping away. That’s why you see foaming cleaners for bath tiles and other surfaces.
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