A strong base like lye sapponifies the fatty ester. This breaks off the glycerine, and the fatty acid combines with the alkali to create soap. The fatty acid will bind with fat soluble compounds (non polar), and the alkali binds with water (polar). This allows the soap to grab dirt and the water rinses it off.

The glycerine left over from soap production is the same glycerine you can buy at the pharmacy as a laxative.