The –idae suffix is a Latin plural counterpart to Greek –idai (singular –idēs), meaning offspring. In the plural, the –idai suffix was used to denote tribes or groupings of people with a common ancestor; e.g. the Heracleidae, the descendants of Hercules. It was also generalised to names of dynasties, and not just Greek ones either: the Fatimid Caliphate is so called as a Hellenisation of al-Fātimiyyūn “offspring of Fatimah”.
So by analogy with Heraclids, “the tribe of Hercules”, and Fatimids “the tribe of Fatimah”, you get felids, the tribe of cats.