Why did Christianity spread throughout the Roman Empire?

Aggressive missionary work was the initial cause for Christianity's spread, followed by mandatory conversion as decreed by Theodosius in 380.
The Roman Empire was ruling the known world, primarily in the West at the time of Jesus Christ. About a dozen years after His death and resurrection, His Apostles began their ministry of spreading and converting all to become followers of the 'Way.' This movement soon became known as 'Christianity.'

In the early 4th Century, the Roman Emperor, Constantine, approved of Christianity and openly allowed worship in it. Near the end of the 4th Century, the Emperor Theodosious I made it the official language of the Empire and shortly thereafter, banned any form of pagan worship. From this point forward, the Roman form of Christianity blossomed from generation to generation.
Saint Paul is credited with spreading the Christian faith among the Gentiles - the non-Jewish population of the Empire. Expansion of the Christian faith was slow at first, but Christianity often appealed to poor people who felt alienated by mainstream religions.

By the beginning of the 4th century CE, Christianity is believed to have converted about ten percent of the population of the Roman Empire. Scholars believe that at this stage Christianity was evenly split between the proto-Catholic-Orthodox faith and Gnostic Christianity.

In the early 4th century, Emperor Constantine gave Christianity state patronage, after which the faith began to expand rapidly, as many felt that it was socially or politically desirable to embrace Christianity. Emperor Theodosius I made Christianity the official state religion of Rome in the 380s, and banned worship at pagan temples in 391. The spread of Christianity was soon complete.

Additional Answer:

Paul went on 3 missionary journeys telling people the good news that Jesus had died for them but rose again.
At first, Christianity spread slowly, mainly in the Greek-speaking east, until Emperor Constantine gave the new religion state patronage and offered various inducements to become Christian. In the end, it was a matter of compulsion. Emperor Theodosius made Christianity the official religion of the Roman Empire in 380 CE, at a time when Christians were still a minority of the overall population of empire, and banned the public worship of the old gods in 391 CE. Persecution of pagans had begun early in the fourth century, under Constantine, but under Theodosius it reached ferocious intensity. It was even a capital offence just to look at a pagan statue that had been smashed by the Christian mobs. Christianity spread throughout the Roman Empire because only the most pious of pagans were prepared to risk life and liberty for their faiths.
Christianity initially spread across the Roman Empire because people were attracted to the message. This happened despite the fact that Christianity met opposition from two quarters:
  • From those Jews who rejected the idea that Jesus should be identified as their Messiah (or Christ), and therefore found the teachings about Jesus quite offensive.
  • From the Greeks and Romans who considered Christianity an unnecessary innovation, and who considered it a threat to either their income or the proper ordering of society.

For Greeks, Christianity had great appeal as it met the highest objectives of their main philosophical schools (Platonic, Stoic, and Aristotelian), without the baggage of the traditional cults, and their gods, and their misdemeanors. This also applied to Romans who had been influenced in this way by Greek culture.

For those on the fringes of the mainstream Greco-Roman culture, the tribes of the lands which had not yet been substantially Hellenised (made Greek speaking), Christianity came to offer easier access to a higher culture than was available through other methods of cultural interchange, since each community tended to follow its own cult.

For poor people, Christianity provided a means of validation of their self-worth, since in Christianity wealth is not the measure of a person. There was also an aspect of mutual help in Christian communities that must have made them vibrant and attractive to others.

These can be considered to be a starting point for Christianity's spread. As time went on political factors also had a part to play, and these are discussed in the related question, listed below, "How did Christianity spread throughout the roman empire?"

1. The apostles travelled around testifying about God`s kingdom and that Messiah had come.
2. They told about Jesus Christ as first hand witnesses and the Holy Spirit was with them.
At first it grew by peaceful means through a vast preaching work. By the 4th century the Roman Empire adopted Christianity as it's official religion and began forcing it on the populace to various degrees throughout history.
The appeal of Christianity. It is generally agreed that Christianity's promise of redemption and salvation was appealing to many Romans. In pagan religions it was thought that after death one went to the underworld and not much was said about the afterlife. Christianity, instead, offered salvation and hope. Moreover, the pagan gods were indifferent to humans and they were gods to be appeased with sacrifices (natural disasters were interpreted as the wrath of the gods because they had not been honoured). By contrast, Jesus Christ was a figure of compassion, mercy and benevolence.

Imperial endorsement.. At the time of Constantine the Great, the first emperor who favoured the Christians, Christianity was influential, but still very much a minority religion. Constantine promoted Christians in the imperial bureaucracy, mediated between competing Christians doctrines, built important Christian churches and enacted laws favouravle to the Chritians.. All but one of the subsequent emperors were Christians. In 380 the co-emperors Theodosius and Gratian issued the Edict of Thessalonica which made mainstream Christianity (the Latin and Greek Churches, which later came to be called Catholic and Orthodox respectively) the sole legitimate religion of the empire. Christianity became state religion ad was officially endorsed by the Roman state. The object of the edict was to ban the many dissident Christian doctrines which were branded heretic. The main target was Arian Christianity, which was popular around the empire and which was persecuted. Thus, the edict also promoted greater uniformity among the Christians.

Persecution of pagan religions. Theodosius also introduced laws which banned the institutions and practices of Roman religion which had not already been banned by previous Christian emperors. This made practicing this religion virtually fully illegal. Theodosius either ordered the destruction of pagan temples, shrines and statues or condoned such destructions by Christian monks and zealots. In some parts of the empire there were also forced conversions. These persecutions weakened pagan religions and further favoured the spread of Christianity.
People needed something to believe in in and at that time people neede to be accepted
Christianity was institutionalized by Constantine I.
Christianity seems to have arrived quite early in Rome. By the fourth-century time of Emperor Constantine it was the only city in western Europe to have a significant proportion of Christians, possibly over ten per cent.

Christianity spread in the Roman Empire because of how much they were persecuted. This may sound weird, but it meant that there were only real Christians, no one who wasnt one claimed to be one. This kept Christianity nice and healthy, unlike another time when you were forced by the Church to be Christian, and this was when Christianity went off tracks for awhile.
Also, people were amazed that Christians would give up their lives so horribly for their God and so the Romans started to think that if they would give up their lives for this 'god' then there must be something real in this faith.

because it appealed to the roman lower class. it's fair to women so women joined in record numbers. slaves liked it because appreciated them as people. and the young church was firmly rooted by the apostles causing it to spread. the final reason was emperor Constantine, he was a Christan and changed the roman religion to Christianity.
Constantine gave Christianity state patronage and ensured that men of ambition knew that their career prospects would be enhanced by appearing to embrace Christianity. The Church was given the opportunity to distribute state food aid to the poor, enabling them to proselytise among them, and the lower classes were encouraged to emulate the elite who professed to be Christians. Constantine also promised every Roman convert a white garment, with twenty pieces of gold. He began the persecution of the pagan temples, continued with even greater intensity by his Christian sons and successors.

There were many good reasons for Christianity to spread in Rome.