The joke is a play on words. Dieters are usually watching their figures and want to prevent fatty foods or carbs from adding weight to parts of their bodies, such as the waist area.  Letting something 'go to waste' means to not use all of something and have to throw away some left overs.  The phrase sounds more like the dieters don't want any fatty foods to add weight to their waists.
The dieters vowed to let nothing go to waist .