Selective breeding was obviously known about for a long time. Darwin himself was quite inspired by it, and the term "natural selection" is actually an adaptation of the "artificial selection" used by breeders (the metaphor is that Nature, with a capital N, is doing the job of the breeder). Darwin in particular was interested in pigeon fanciers, who could breed birds that not only looked quite different from others (and make those traits "breed true"), but also acted differently (e.g. "tumblers" that fly in a very erratic pattern), showing the power of the art of selection.

But, it should be clear, nobody — not even people who believed in fixity of species — would ever have denied that when you breed two organisms together, the third is some kind of hybrid of the other two. This is obvious to anyone who has looked at family resemblances. The children of two short people are often short; the children of two tall people are often tall. (But not always — this "blending" mode does not take into account what we now know as Mendelian characteristics, which don't act in this way, and are responsible for what were called "sports" in Darwin's day — the occasional black sheep, the albino, the sudden change.)

The question was whether or not such blending can lead to a different species. E.g., could you, with enough breeding of dogs, end up with a cat? Or even just a dog that could not reproduce with another dog and produce fertile offspring, to use the species concept that was fairly established by Darwin's day (per Buffon).

Keep in mind that they did not have anything like a modern understanding of heredity. Not just not understanding DNA and chromosomes, but not even knowing Mendelian characteristics or any sense of there being a real "unit" of genetic information (the term "gene" was not coined until 1909; Mendel's work was contemporaneous with Darwin but was little-observed and Mendel thought it only applied to plant varieties and that it was evidence again evolution, because it is just reshuffling of existing information and not the introduction of new information, so it wasn't going to be part of this conversation).

So what someone like Darwin could offer up was a vagueness where he said, "well, sure, if you kept breeding dogs for specific traits for a VERY long time, it might end up a different species." He had no way to prove this; he didn't really have a great access to the fossil record, and so a lot of his arguments were more along the lines of, "look, these different species seem like they are related, so maybe they diverged from a common ancestor that I posit must have existed." That is kind of a weak argument in terms of the evidence, even though it turned out to be correct! It also turns out that it was totally unclear whether the Earth and life were old-enough to support the VERY long span of time that would be needed to turn, say, a worm-like creature into a man-like creature. Forget dogs into cats — try to turn worms into men! You can see why a lot of very educated people thought, "I just don't see how that can work out — it is less of a leap to assume that some sort of supernatural power was at work." Even today it is rather remarkable to imagine how you end up with different species that have, for example, totally different numbers of chromosomes and genes (the answer is the Darwinian one: very, very gradually) which ought to be wholly incompatible with one another and thus apparently preclude the idea of "intermediate" forms (but genetics turns out to be super complicated).

So you can see that it is actually quite a big leap to go from "selection can turn dogs into different breeds (who are all the same species)" to "selection can turn dogs into not-dogs, and in fact is responsible for all life having a common ancestor some unfathomable distance into the past." And I say unfathomable in all seriousness: even Darwin did not consider that the Earth might be billions of years old (in his day, positing that it was several hundred million years seemed like quite an imaginative leap). We take these numbers for granted today, because we have come up with very good ways of showing them to be real and have been introduced to these concepts of "deep time" since childhood, but they are conceptually quite large leaps from what was understood prior to the early 20th century.