Why did Rome manage to conquer the Eastern Mediterranean?

The main factor of success is the military organization of the Romans (both during the period of the republic and during the empire), division into legions, a large fleet for the transfer of troops by sea. Already in 14 AD, the empire was home to about 4 million people capable of wielding weapons. The army included several dozen legions scattered across the provinces, many siege machines. Plus diplomacy, especially under Emperor Octavian Augustus, during which contacts even began with China.
No other state in the eastern Mediterranean had such power.