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I'd just do it in exposition; not a lot of it. Modified in response to clarifying comment of OP There are two scenarios; based on your comment. Either the characters know they used to be humans, ...
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#4: Attribution notice removed
Source: https://writers.stackexchange.com/a/38719 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#3: Attribution notice added
Source: https://writers.stackexchange.com/a/38719 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#2: Initial revision
I'd just do it in exposition; not a lot of it. **Modified in response to clarifying comment of OP** There are two scenarios; based on your comment. Either the characters know they used to be humans, or they don't. Since our own species (homo sapiens sapiens) existed 50,000 years ago (at least) and we don't really know anything about our ancestors or what they suffered through 35,000 years ago, it is plausible the current residents don't think anything about having been humans at one time, and may not even know that they were. In that case, I'd just introduce your Weasel as a weasel, he sees his friends and business acquaintances as whatever animals they are. Don't explain, this is just the normal world. Most stories open on the "normal world" of the MC, and you should present that world as they experience it. He has no reaction to the store clerk being a 6' porcupine or the cop being a 4'5" hedgehog; that is the way of the world. Now if they DO know they were once human and are now animals, then in the opening have your MC doing something that allows his mind to wander, like walking somewhere, and reflect for a sentence or two on the transformation. Perhaps he wonders what it would be like to be human, as he has heard the Atlanteans are, instead of a weasel. He wonders if it would be any different; they certainly looked colder without any fur, and more prone to injury. And he has heard they have the weakest sense of smell, he couldn't imagine going nose blind. On balance, he was happy being a weasel. Weasels were rare, it gave him a certain measure of recognition in society. Something like that. Not **straight up** exposition, exposition filtered through some thought and feeling to create conflict: He is supposedly cursed but prefers his cursed state, it is all he has ever known, and changing from weasel to something else is as undesirable as us being forced to change form into some other animal.