Back then there was no regulation on transportation. Usually they brought animals (such as pigs, cows, goats, chickens, ect) as food, but back then animals always found their way on a ship. Some people would also bring animals as pets or experiments.
The typical excuse for their overpopulation is that these animals manage to escape. They'll start their own communities at the cost of disturbing the environment. Nowadays it's very hard to bring plants and animals overseas, but it still happens sometimes. The biggest disturbances today, however, started between the 16th-19th century.