There are many aspects regarding how did Christianity spread out in the World. Perhaps we might have similar reasons why did Muslims spread out as well. Both religions have their foundations in Judaism, so they are kind of "Judaic Sects", although at their beginning Muslims consider to be a Christian Sect.
What did prevented Judaism to spread out in the Ancient World? I guess it was the fact that their God, was God of wrath with laws like, "an eye for an eye". Besides the Hebrew faith was based upon Mesopotamian legends and stories like Noah's Flood, a reproduction of Gilgamesh Epic and even the Mosaic Law is a copy of the Hammurabi's Code, and that is not a surprise to have many similarities, since Abraham came from Mesopotamia region and moved to Palestine.
Then we have the same pattern of Jesus story known to the Greco-Roman World. Mithras a Roman God, He was born through a Virgin Mother, had 12 disciples, he was considered the Light of the World, the Savior, died and resurrected for the sins of mankind. The same pattern is founded with the Egyptian God Horus, Greek semi- God Adonis. Again the same story is repeated through the Hindu Krishna (Christ) and Buddha.
So Christianity, or Jesus Christ story and teachings to be more accurate, are not something new to the World. They existed in different variations in many places, that's why it was easy to be accepted.
A key role, and one of the major factors of Christianity's expansion was the Hellenic Culture domination of the Ancient World. Since Alexander the Great conquered the known World he passed the Hellenic language, religion and philosophy to many other places. Socrates, Plato, have similar ideas and teachings like Jesus did, so for a "new" religion to appeared wasn't hard, since it had the same accepted principals. So we have the unique Hellenic Culture all over the Roman Empire from Minor Asia, Middle East, Black Sea, Balkans, Africa(just to remind you Alexandria and the vast Library it was build by the Greeks) Europe and Mediterranean Sea to be the dominated by the one and only Hellenic Culture and Language. Besides Jesus preaching was more successful to the Hellenic speaking Jews, instead of the native speaking Jews.
At the early stages of Christianity, the first 300 years it was persecuted by the dominant Roman religion of the era. So if you persecute something, it usually thrives. Christians had a unique bond -Love that was Jesus fundamental teaching. Jesus preaching gave an update role to women, we all know from the Gospels the way He spoke and deled with women, plus He first appeared to them after His resurrection from the death. Even in His Mother Womb we have the prophesy that "she will be honored like no other Woman on earth" and this honor it is clearly seen within the Orthodox/Catholic Church.
The emphasis on women appeared in the Gnostic Gospels and especially in The Gospel of Maria Magdalene, a gospel that was banned from the Biblical Canon (along with some others like the Gospel Of Thomas). We know that, at the Early Christian Church there wasn't any structure or central control, regarding of what Books could the congregations read, besides the 51 books of the Old Testament. We had many other writings circulating along with Epistles of the Apostles and along with them they used to be the Gnostic Gospels. It seems that in the Early Church women played an equal role like men, and this is obvious in the Gospel of Magdalene, that women could serve in the Christian Church. We even find early paintings in Rome's Catacombs that women served as priests. There was a conflict for the role of women within the Early Church and this conflict ended when Constantine declared Christianity as the official Religion of the East and West Romanae. The structure started to be more official and Church Councils (Synodos) started to take place in order to clarify Christian Faith, starting with Council of Nicene in 325A.D with the Trinity issue. Then the Biblical Cannon was clarified by Saint Athanasius in 365A.D. and sealed for good the Books that consisted the Canon, leading to fade the Gnostic Gospels.
So we have many reasons why did Christianity spread so fast:
1. The common Hellenic Culture and Language amongst the Greco-Roman Empire
2. Jesus message and teachings that was Universal to all
3. The important role of women in the Early Church
4. Constantine the Great who turned Christianity the Official Religion of East and West Roman Empire
