But if you added an equal amount of clear water, that coffee would become more transparent. In a similar way, by dissolving soap into a solvent, you increase the clarity of that soap. Second, there are certain techniques to making transparent soap which involve the manufacturing process itself. Again, Catherine's book discusses this in detail. Catherine's soap relies to a large degree on grain alcohol to make it clear and remeltable.
Grain alcohol is both expensive and hazardous to work with.
While in theory, one could make a clear, remeltable soap using glycerin and sugar water only, the resulting soap would be of very low quality. Manufacturers have therefore developed Melt and Pour soap bases which substitute the grain alcohol with various other options. A very common substitute is Propolyene Glycol. Propolyene Glycol is a petrochemical which is commonly found in candy coloring, shampoos and many commercial bath and body care products. However, there are many other chemicals which may be used to make the soap clear and remeltable.