Yeah, I caught parts of SWAT Kats as a kid, too.

As for "how this": good 'ol dimensional shenanigans, probably. Normally I don't like the "human turns into an anthro" trope, but maybe one of the SK villains pulls that on Vince and Durandal in an attempt to force the duo to work with them (since if they go back home, they're stuck like that)? Naturally they refuse, and find themselves at T-Bone and Razor's place in short order. I did doodle Durandal as a cat person, but I'm not quite sure how Vince would look yet (though that's not why he's still human here--I wasn't thinking of furry-ifying going in); maybe that's for someone more familiar with the show's design language to figure out? Also, I don't remember seeing episodes with Felina as a kid, but she definitely feels like the "here lies the root of my furry-ness!" type.

Incidentally--one hopes that transformation spell dulled Vince's cat allergies.