Eugenics as an approach to "bettering the human race" through the science of heredity had a very broad political appeal, in particular as it could take a wide variety of forms. Today we generally associate "eugenics" with the most coercive practices done in its name — compulsory sterilization, immigration restriction, anti-miscegenation laws, even genocide — but under its banner also were many relatively innocuous practices like birth control, science education, pedigree tracing, what we today call "genetic counseling," and even things that have only a tenuous connection to heredity.

People who were on the leftish side of the political spectrum (e.g. J.B.S. Haldane, Margaret Sanger, W.E.B. DuBois) liked its interventionist, improvement, and rationalist pretensions. People who were on the rightish side of the political spectrum (e.g. Madison Grant, David Starr Jordan) liked its Social Darwinist, judging sorts of aspects. My reading of the works of both sides has made me feel that they did not basically see eye-to-eye despite their apparent embrace of the same approach; the leftish/socialist/etc. eugenicists had really different preoccupations and methods than the conservative eugenicists. It is probably worth noting that the only bloc that nearly always opposed eugenics were the Catholics, who found too much in it that conflicted with their views on reproduction. It is also worth noting that the specific commitments of people who considered themselves proponents of "eugenics" varied a lot — some were very vague, some saw it merely as a form of "study," some wanted to sterilize people by the hundreds of thousands.

Eugenics also appealed to people who wanted to invoke a secular, "scientific" basis for the ordering of society and improvement of the world. Mendelian inheritance had just been "re-discovered" and given a strong physical basis with the work of T.H. Morgan on chromosomes. In its simplest forms it seemed like it would be relatively easy to identify the genetic traits that were responsible for many "bad" aspects of human existence and breed them out, in analogy to how animals are bred for specific traits. That human genetics was vastly more complicated than this, and that the nature-nurture divide is a complex, dynamic interaction rather than a simple flow chart, and that even in the case of simple Mendelian recessives, "breeding them out" is a non-trivial endeavor (as the Hardy–Weinberg principle made quantitatively clear), was easily ignored by popularizers. Even those biologists who knew it was more complicated than this generally supported the idea of eugenics as a form of applied biology. (That eugenics is still quite popular today amongst a certain educated crowd testifies to the way in which a little hope and/or hate can outweigh a lot of sound biological knowledge.)

Eugenics only started to fall out of favor when it became primarily associated with conservative cranks who thought that sterilization and the like were the answer to everything and produced blatantly substandard scientific studies to back up their political predispositions. (There were other factors that went into the changing of American public opinion as well, including the Great Depression, the push back against the pseudoscience propagated by the Nazis, and the public rise of cultural anthropology and the subsequent propaganda efforts against racism of many of its prominent supporters, including in the newly formed UNESCO organization.) The scientists who studied and applied hereditary issues distanced themselves fairly early on, and some of the new fields took pains to explicitly move away from the term "eugenics" ("genetic counseling" was coined for this explicit reason). It was not until the 1960s and later that the association of eugenics with Nazism took on its full cultural force and the term "eugenics" took on its primarily coercive and pejorative bent that it has today.

In an article I wrote about California's sterilization policies, I quote one of my favorite little quotes about the difficulty of look at the term "eugenics" from a historical viewpoint:

Historians of science, however, have found that the term is chameleon-like, changing definition, purpose, scope, and values in different eras, countries, and social settings. At one extreme, eugenics is a gigantic umbrella that covers almost all social movements in which sex, gender, heredity, family planning, reproductive options, marriage, immigration, social status, and social failure are involved. It ranges from concerns about the most dependent children and adults to interest in the most successful and eminent high achievers and their roles in shaping future generations of humanity. At a more restricted level of historical interpretation, eugenics is the application of human heredity to an analysis of differential birthrates. The broader historical approach makes eugenics a more difficult target for those concerned about personal liberties. The narrower approach makes the old-line eugenics of the first half of the twentieth century a dead horse that is no cause for present worry.

I tend to favor today using the more narrow approach to talking about "eugenics," which means that many of the early fans of eugenics were not much of eugenicists at all. This is an analyst position, though. If instead you decide that "eugenics" means everything that people have used it for in the past and present, it (to quote myself) "has been applied to everything from applied to practices from expectant mothers taking vitamins for the health of the fetus to the worst atrocities of Nazi genocide," which in my opinion makes it uselessly empty. However that emptiness is partially the answer to your question — it meant many, many different things to many different people in the past, and for almost none of them did it mean actual genocide, and for only some of them (though it is a large "some") did it mean coercive practices.

Remember, the word "progressive" does not inherently infer a specific path of progress or a set of values that drives the "progressive" agenda. To be a progressive circa 1914, one would hope to help progress society towards some ideal through means, measures, and policies that you think think will be effective, but you may be very far from agreement with both progressives of other time periods, and others from your own time period who may be also progressive but with a different set of values, a different end goal, and different means in mind.

It is only the belief in progress that encompasses all those who were progressive, which was in contrast to either conservatism or reactionary ideologies that wished to either maintain the status quo or revert to a past social order. Progressivism merely stipulates that advances in social and economic organization and science and technology can lead to the betterment of human societies and the human condition.

Towards the end of the 19th century in the beginning of the 20th, scientific progress was on the move and it was popular to believe that scientific rationalism could be applies to public policy and social organization to transform society in a positive way. Eugenics was pseudoscience, and we have known it to be such for a long time now, but in the era we are discussing, it at least seemed like it was scientific to many people who had been influenced by the racism and classism of the time and who found the idea of applying (their misunderstanding of) the theory of natural selection to the human species and to take the reigns from nature through public policy regulating human breeding. Since Eugenics as policy is inherently progressive (as in, transformative), anyone who favored eugenics as policy was in some way progressive, but there were plenty of people who also believed in the social benefits of scientific progress and scientific rationalism that disagreed with eugenics and saw through the psuedoscientific facade of it.

Coming from a biotech background rather than a historical background here.

One hundred years ago, the foundations of modern genetics had just been laid. Basically everything a student learns today in middle-school and high school biology class about genetics either had been discovered in the past ~10 years or had not been discovered yet (remember, the structure of a DNA molecule wasn't discovered until the 1950s). Here's a pretty decent timeline for some specifics.

So, imagine for a second that there's an upheaval in science. Someone discovers that the genetic material for every trait is passed onto a creature's offspring independent of all other traits. Seemingly every trait in a pea plant can easily be bred out of a population. Want tall pea plants? Discard the short pea plants or at least keep them from breeding. Want green peas? Discard the yellow pea plants or at least keep them from breeding.

Now, consider for a moment the implications if all human traits and behaviors were inherently genetic. If you could eliminate the "alcoholism gene" from the gene pool, no one is ever killed by a drunk driver again. No child is ever beaten by a drunk parent again. If you could eliminate the "rape gene" from the gene pool, no one is ever raped again. If you could eliminate the "cancer" gene, "murder" gene, "stupid" gene, even the "unattractive" and "lazy" gene, everyone could live healthy, long, productive lives. Just consider again: If I could scientifically guarantee you that in a couple generations, no one would ever be raped again, would you go for it?

Now take into account casual attitudes about racism at the time. If a historian specializing in this time period would like to chime in with a more informed opinion, I'd appreciate it, but very brief research into European/American attitudes towards blacks or Asians (or just American attitudes towards Irish and Italians) should prove fruitful quickly.

So if you take a populations of people that is used to saying "Group A is better than Group B," and combine it with incomplete science that says, "Eliminating Group B can eliminate all societal ills," you get the foundation of eugenics.

Another important part of the discourse I'm not qualified in the least to talk about is modern philosophy. I think a discussion of 1900s notions of truth and value would add a lot to understanding the mindset of progressives at the time.