First, the Empire united much of the ancient world under one rule. This made it easier for Christians to travel and spread their message, and it also created a common language, Greek (and eventually Latin), that could be used to communicate the gospel.
Second, the Roman Empire provided a degree of stability and security that allowed Christianity to grow. For centuries, the Empire was a relatively peaceful and prosperous place, which gave Christians the opportunity to build churches, schools, and monasteries, and to spread their faith.
Third, the Roman Empire unwittingly provided Christianity with a powerful new symbol. The cross was originally a Roman instrument of execution, but Christians began to use it as a symbol of their faith, and it quickly became one of the most recognizable symbols in the world.
On the other hand, the Roman Empire did certainly persecute Christians at times. The persecution of Christians began in earnest under the emperor Nero in the 1st century, and it continued intermittently for several centuries. Christians were often accused of being cannibals, atheists, or enemies of the state, and they were subjected to a variety of tortures and executions, including decapitation, crucifixion, and burning at the stake.
Despite the persecution, the Christian faith continued to grow and spread throughout the Roman Empire. By the 4th century, it had become the majority religion in the Empire.