Lots of people love anthropomorphic characters. Perhaps you are one such fan. Perhaps you are a writer who plans to feature them in your fiction. Many authors don’t feel a particular need to justify anthropomorphic characters’ presence in their stories. There are plenty of examples available, but attempting to list all the relevant folktale figures, manga characters, and inhabitants of Duckburg would take up an entire essay, at least. But there are other people—people like me—who become anxious if important elements aren’t given a backstory or explanation. For those people, here are some semi-plausible ways anthropomorphic characters could have appeared in your setting…
There’s the surgical approach: Doctor Moreau, for example, was quite keen on surgically sculpting animals into forms that he considered more pleasing. This effort did not entirely go according to plan, but still, it was an impressive result for someone limited to Victorian medical technology. Now passé.
Drugs: Take Jack Kirby’s comic book character Kamandi. A scientist developed a drug which he hoped would enhance animal intelligence. There’s no hint that he intended for the enhanced animals to start ambulating on their hind legs and imitating various human cultures, but that’s precisely what happened after the Great Disaster wiped out most humans.
Genetic engineering and other forms of biological manipulation: S. Andrew Swan’s Moreau series features engineered anthropomorphs. Cordwainer Smith imagined Underpeople and Masamune Shirow imagined the Puma sisters. Then of course, there’s Brin’s Uplift series. Why create anthropomorphs? In the Uplift series, it’s for galactic status. In other series, it’s often because humans want expendable others to do the heavy lifting. Or exploitable beings to use for unsavoury purposes. Note that if the creator humans are still thriving, it’s probable that the uplifted animals lead unpleasant lives.
Sometimes the humans are gone. Revolution! Planet of the Apes! But in most cases it seems that humans killed themselves off and other animals took over their ecological niche. Given enough time, the new species could become sorta kinda humanoid (See Stableford’s Realms of Tartarus. Granted, humans still existed in that setting, just not where the new intelligent species were evolving.)
Alternatively, if you don’t want to set your story in the sufficiently distant future, there’s always another trouser-leg of time. Perhaps humans never evolved at all, leaving that niche entirely open for another species to fill. (See Boyett’s The Architect of Sleep)
Or you could always start with humans rather than animals. Lots of humans like to dress up as animals, with some opting for more permanent alterations. SF authors have also imagined ways to do this: see Tanith Lee’s Don’t Bite the Sun or Charles Sheffield’s Sight of Proteus.
Why not start with aliens! Surely, somewhere on the millions of inhabitable planets out there (cough) species have evolved to look like Earth species with a dash of human. That’s how you get lion folk (Chanur’s Hani), ant folk (Serpent’s Reach’s Majat), or bear folk (Spacial Delivery’s Dilbians). Just set the book on an alien world and practice saying, “convergent evolution.”
But the easiest way to introduce anthropomorphs to SF is just to write them, put them in spaceships, and eschew obsessing about backstory: Here are catpeople! Readers won’t really mind as long as the story is interesting.
In the words of Wikipedia editor TexasAndroid, prolific book reviewer and perennial Darwin Award nominee James Davis Nicoll is of “questionable notability.” His work has appeared in Publishers Weekly and Romantic Times as well as on his own websites, James Nicoll Reviews and Young People Read Old SFF (where he is assisted by editor Karen Lofstrom and web person Adrienne L. Travis). He was a finalist for the 2019 Best Fan Writer Hugo Award, and is surprisingly flammable.