Christianity spread because it was attractive. It provided an alternative to the polytheistic status quo, and it's always nice to feel you're on the cutting edge. The new idea is always the better one. It also allowed people to keep many of the holidays and traditions they liked, just with some Christian twists. People like their holidays and they like gradual change in their daily lives.
Christianity is one of the three missionary religions; the others being Islam and Buddhism. All three have morphed themselves at times to meet the needs of particular groups and cultures. The success of Christianity, however was its audience. Christianity appealed to the poor and unfortunate in society. Jesus in one of his first Sermons said
The spirit of God is upon me, to teach good news to the poor.
During his ministry, Jesus often ministered to the sick and poor. It offered hope to those who had none.
A second reason was its acceptance within the Roman Empire. Interest in the old Greco-Roman gods had declined; and with its popularity among the poor, the emperor Constantine for purely practical reasons legalized the religion. Because of its popularity it soon became the official religion of the empire, and spread throughout it. It is no coincidence that after the decline of Rome, Europe was known as Christendom.
One way to answer this is to say that it was able to spread because it was willing to make changes in order to convert people. It was willing, for example, to do things like having Christmas fall at the beginning of winter so as to give Christians a chance to have a major festival when their pagan friends were having one. In various places, it was willing to accomodate local beliefs by letting local people equate local spirits with various saints. By doing things like that Christianity made itself acceptable to more people.
I'd like to see this moved to the Religion Discussion Forum so you can get input from others. I'm guessing none of us will identify all of the many contributing reasons.
The easiest person to identify with the spread of Christianity is Paul. His travels throughout much of the Mediterranean Sea area enabled him to spread the new faith to many locales. Paul was highly educated and fluent in the languages and customs of the many cultures in that part of the world. This allowed him to tailor his messages to the particular needs and situations of the audiences he addressed. He must have been fairly charismatic because he was able to attract large numbers of curious people in many locations; he certainly was able to communicate persuasively and to convince others of the validity of the faith he was promoting. The life he led and the way in which he related to those around him also demonstrated the values of the new faith, which attracted many who were searching or receptive to a new way of relating to people and God.