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The dregs, in *some* beers, is yeast. For beer that is bottle-conditioned [i.e. naturally carbonated in the bottle], it requires that there be yeast present and a little added sugar at bottling time and the yeast uses that sugar to create the CO2 that carbonates the beer. Some breweries use a different strain of yeast for bottle-conditioning, but most tend to use the same yeast that they used for fermentation.

Many breweries use "house" strains of yeast, that are not commercially available to homebrewers through any yeast producers. The dregs give you access to that strain.

So by saving the dregs, and culturing that up into larger quantities of yeast by feeding it, you can have access to the same yeast a commercial brewery uses--important if you want to make a beer either that is a clone of something they produce or just really like the flavor profile their yeast provides.