Why did many Christians flee ancient Rome and move to Axum?

To avoid persecution.

Explanation:

The Kingdom of Axum was a monarchy located in the territories of present-day Ethiopia, Eritrea and Arabia, which existed from the 1st century BC to the 7th century AD.

This nation was one of the first to adopt Christianity as an official religion, specifically in 325 AD. At that time, many Christians from other nearby kingdoms and nations, including the Roman Empire, began moving to Axum, which guaranteed as a state the security and free celebration of that religion.

Rome, for the same time, had issued the Edict of Milan in 313, which did not guarantee Christianity as an official religion, but the safety of its believers. Christianity became the official religion of Rome only in 380 AD.