They're mostly the descendants of backyard chickens that got loose, and Grand Cayman is an easy place for a chicken to survive.
Keeping chickens was a normal part of Caymanian life for generations. Before tourism and offshore finance transformed the island from the 1970s onward, Cayman was a small, fairly self-sufficient place of fishermen and seafarers, and plenty of households let their chickens roam the yard. Some of those birds were always wandering around.
The big jump is usually blamed on Hurricane Ivan in 2004. Ivan hit Grand Cayman hard and wrecked homes, fences and coops across the island, and a lot of domestic chickens escaped or were abandoned. Once they were out, they bred with the free-roaming birds already there, and the population took off. Kauai in Hawaii went through something similar after Hurricane Iniki in 1992, which is why it's also famous for feral chickens.
A few things have kept the numbers high since then:
Few natural predators. Grand Cayman has no foxes, mongooses or large wild cats, so not much hunts adult chickens beyond the occasional dog or cat.
Year-round breeding. The warm climate means hens can raise chicks in every season instead of stopping for winter.
Plenty of food. Beaches, restaurants, bins and tourists dropping or deliberately sharing snacks give them an easy living, which is why they gather where people eat and hang out, like Seven Mile Beach and George Town.
Locals have mixed feelings about them. Some see the chickens as part of the island's character, but many find them a nuisance: roosters crowing at 4am, birds scratching up gardens, droppings. The government and private contractors have run culling and trapping programmes at various points, but the chickens breed fast enough that the population keeps bouncing back.
So the chickens aren't a sign the place is less modern than it looks. They're leftovers from its rural past, given a big boost by one very destructive hurricane.