1. Environment. Because the Fertile Crescent has got 2 rivers(Tigris and Euphrates), and water is a crucial thing for plants and animals to grow. If many of them grow, they then spread.
2. Trade. The pastoral society from the Fertile Crescent have successfully brought different sorts of animals to other regions through trades and transactions.
3. Tech. After the agricultural revolution, people started inventing different advanced tech, such as plows and wheels. These technologies have made the growing of crops and animals become more efficient. As the production came up, they've got more chances to spread.
4. Migration. Groups of people who have migrated from Fertile Crescent to other places had brought their own crops and domesticated animals, which had rapidly causing spread of these things.