The two most important factors that helped the alphabet to spread among these cultures were trade and conquest.
The Phoenicians were a trading society that originated in the eastern Mediterranean. As they traded, they exposed other cultures to their system of writing. One of these was the Greek culture. The Greeks modified the alphabet and then spread it as they colonized. Through contact with the Greek colonies (some of which involved conquering them), the Romans came to use this alphabet. They spread it across Europe through conquest. The people to whom it spread then conquered places like the Americas and brought it with them.
Most interactions between cultures in ancient times happened via trade and/or conquest. The spread of the alphabet fit this pattern.