Was it too disorganized? An early example of nonviolent resistance, succeeding in the same way and for the same reasons as Gandhi and King? Or were the causes too complex to summarize in a Reddit comment and require an entire book to answer?
The persecution of Catholics in Rome ostensibly began with Nero attempting to redirect the rumor that he had started the Fire of AD 64 in order to build the Domus Aurea. After this the persecutions continued sporadically and many of them ineffectually simply because the Romans could not be bothered to seek out this miniscule population of their empire who were peaceful and non bothersome save for their refusal to serve in the military. This is not to say they were trusted and ignored if they were open about their religion, however in his letter to Pliny, Trajan says that "they are not to be sought out" and "anonymously posted accusations out to have no place in an y prosecution." With this is shows that the atitude of emperors in the 2nd century were more lax towards christians. When Pliny discusses the executions he carried out of Christians he almost makes it seem they were executed for obstinancy more than their religion.
I interrogated these as to whether they were Christians; those who confessed I interrogated a second and a third time, threatening them with punishment; those who persisted I ordered executed. For I had no doubt that, whatever the nature of their creed, stubbornness and inflexible obstinacy surely deserve to be punished.
Simply put the Empire was too large and Christians, as a denomination, too small to be of much concern for later emperors. Pliny also states that Christians were of little to no threat as their services contained little of poltical consequence as well as the participants compliance with roman law.
They asserted, however, that the sum and substance of their fault or error had been that they were accustomed to meet on a fixed day before dawn and sing responsively a hymn to Christ as to a god, and to bind themselves by oath, not to some crime, but not to commit fraud, theft, or adultery, not falsify their trust, nor to refuse to return a trust when called upon to do so. When this was over, it was their custom to depart and to assemble again to partake of food--but ordinary and innocent food. Even this, they affirmed, they had ceased to do after my edict by which, in accordance with your instructions, I had forbidden political associations. Accordingly, I judged it all the more necessary to find out what the truth was by torturing two female slaves who were called deaconesses. But I discovered nothing else but depraved, excessive superstition.
Under Trajan and Hadrian Christians were given more leeway in the criminality of their religion and as a result the persecution of Christians slowed to a few individuals drawing accusations against another which were brought before a provincial governor.
As to why it became the Imperial religion I am unsure of the fine details however the belief which often is put forward is that after Constantine, Christianity was Romanized to become more palatable to the Roman population. As a result, more elites adopted it, which then trickled down to the remaining population.