The Hybridverse (or the Hybrid AU, as it's more commonly known) is a popular alternate universe trope within the fanfic and fanart community, where characters are portrayed as hybrids—part human and part animal. These characters typically retain their human intelligence, but possess animal characteristics like ears, tails, wings or enhanced senses typical to their animal counterpart.
Along with possessing the animal characteristics of their animal counterparts, hybrids also often exhibit the traits of their animal counterparts like their instincts (hissing, purring, snarling), dietary restrictions (like bunnies typically being vegetarians or snakes typically being carnivorous) or specific habitat preferences.
This trope frequently explores themes of ownership (typically a hybrid being adopted from some kind of hybrid adoption center), caretaking (typically portrayed as a stray finding their forever home with a human partner). In some universes, hybrids are also ranked with said ranks being based on rarity (with D-rank hybrids typically being the most common and Z-rank hybrids being the most mythical).
There is no specific foundational rules of the hybridverse as so many authors and readers typically have their own interpretation on this AU trope, but typically how a hybrid acts and interacts with the world is based on whatever animal they’re a hybrid of (i.e. a husky hybrid being loud and prone to seeking attention or a snake hybrid preferring to sunbathe).
As for why hybrids exist in the world, again, that's entirely dependent on the author and their worldbuilding. They could’ve been genetically engineered because of some kind of secret government project, where human DNA was combined with animal DNA and that was how hybrids came to be.
Personally I prefer the concept that hybrids have always existed naturally, somewhere down the line of evolution hybrids went in another direction than their animal counterparts, becoming more human, yet still retaining their animal traits.
Other concepts mention hybrids being borne from some religious sect, other concepts mention magic and other concepts simply label hybrids as “completely unexplained”.
Typically hybrids in this universe can’t actually shift into the animal they’re a hybrid, but there are many interpretations of the hybrid AU where the hybrids are able to shift between their full animal form, their hybrid form and their human form (sometimes as a way to hide their true form).
The hybrid AU also differs from the “Furry” fandom, because the characters are generally considered more human-centric rather than anthropomorphic animals, though again this is dependent on the specific author’s interpretation of this AU trope. Some authors might prefer more human-centric hybrids and other authors might prefer more anthropomorphic hybrids.
However you interpret the hybrid AU, is up to you, because again creativity is a versatile thing and authors (and readers) have the free will to interpret and worldbuild this trope in whichever way that works for them.