Breeders used what is sometimes called "folk heredity": they understood that you could blend traits and that if you did it enough, especially with a lot of inbreeding, they would "stick" and "breed true," albeit sometimes they would create an occasional "sport" (something unexpected or un-obvious, like the famed black sheep from white parents). They understood that with time, patience, and luck, you could get a lot of "stable" results (e.g. not just one-off animals but animals that had the same trait and passed them down to their offspring). The basic model was that of "blending" (children are kind of like a mix of their parents) but they knew that this was not a guaranteed thing (and that some mechanisms, like in-breeding, tended to cause traits to "stick" better between generations, without understanding why).

You don't need Darwin to do this kind of breeding. Darwin in fact gives you almost no tools for this on his own, and adds nothing to the explanatory value of the above (Darwin's own theory of heredity was not very popular and had obvious problems with it). The breeders did not have what we would today call a sophisticated theory of genetics — they were doing it based on practice and experience, not theoretical models. Darwin got more from breeders than he gave back to them, intellectually.

Mendelian genetics does, however, because it illuminates many of the "non-blending" phenomena (like the "sports" and why inbreeding matters so much) and once it was "re-discovered" there were many attempts to apply its understanding to animal husbandry. This allowed, for example, for very elaborate genetic creations, like breeding chickens (generation A) whose children (generation B) would produce lots and lots of identical chickens for slaughter (generation C). This sort of complexity requires an understanding of Mendelian genetics to pull off, and could be done by the 1920s. The economic advantage of the described arrangement is that you can sell the intermediary chickens (generation B) to chicken farmers without giving them the ability to endlessly produce more chicken breeders after that generation dies (generation A would be kept under lock and key by the breeders). Very clever stuff, esp. given how rudimentary the understanding of genetics was at the time.

For further reading on the history of genetics that talks a lot about folk genetics, Bowler's The Mendelian Revolution is great.