Large lions need a lot of prey in order to get enough to eat. You only need to be large if there are large prey animals that can only be brought down by a large predator.
I'm guessing that packs of humans armed with spears and arrows killed off the larger prey animals, and the lions starved. Evidence exists that the same happened in the Americans, with the extinction of large American animals happening about the same time that humans crossed over from Asia.
Socialized medicine killed the lions.
Its time to trot out my personal theory on the extinction of megafauna: Note that Aleuts in skin kayaks killed 50 ton whales with harpoons and atlatls. The trick was they made poison (aconite) out of monkshood flower roots for the harpoons. Using atlatls and similar poisons, it would be relatively easy to bring down megafauna, e.g., mammoths who travel in herds. Human tribes would prosper and multiply until the megafauna were extinct. Smaller prey, e.g., deer, would survive.