How did Christianity change the Roman Empire?
This is a great question.
First, Christianity was able to change the Roman Empire through perseverance. If Christians did not really believe in what they professed, they would have been stamped out long ago. Christians in the Roman Empire were willing to suffer and suffer they did in various persecutions. In time, their faithfulness paid off. They gained more converts and in time even the emperor became a Christian, Constantine.
Second, from a psychological point of view, Christianity offered an answer in an age of anxiety. Their otherworldly worldview helped people to see that there was something more than the material world. Other cults also fared well on account of this point.
Third, from a sociological point of view, Christianity was able to give people more room for social honor. For example, women had greater social capital within Christianity. Also people who experienced what sociologists call social dissonance were able to find social status. In Christianity, people found a way to have a new identity.