Grease and oil are nonpolar and insoluble in water. When soap and soiling oils are mixed, the nonpolar hydrocarbon portion of the micelles break up the nonpolar oil molecules. A different type of micelle then forms, with nonpolar soiling molecules in the center. Thus, grease and oil and the 'dirt' attached to them are caught inside the micelle and can be rinsed away. (About.com)

Using micelles for cleaning is not a new idea. Common household detergents contain micelles, which dramatically increase the capacity of the washwater to pick up organic contaminants. Micelles could likewise be used to remove organic contaminants from industrial waste streams. (Massachusetts Institute of Technology)

In a micelle, the tails of the soap molecules are oriented toward and into the grease, while the heads face outward into the water, resulting in an emulsion of soapy grease particles suspended in the water. (Chemistryexplained.com/)