I'm aware that there was a lot of suspicion in the empire around Christianity in the early days, but what was it about the religion which was so enticing to average working Romans in Greece, Egypt, Syria, and Italy that they would feel compelled to abandon the Hellenic faith they'd known for millennia? It must have been really something to gain converts in the face of early prosecution.
I've just been reading Bart Ehrman's The Triumph of Christianity which asks precisely this question (amongst others). He talks about the importance of it being evangelical and exclusive which I agree are vital but also asks precisely this 'why would they convert in the first place' question. His top answers in brief are
The first is why you take it seriously, the latter gives urgency.
One thing which I think conflicts with both /u/Syria_War and /u/MrCitrusfrugt is that he argues against the image of Christianity spread by missions of charismatic preachers (the 'travelling salesman' model mentioned). He sees Paul as more or less unique in travelling around converting people and says you don't see similar at his time or indeed until perhaps 4th century. Instead Ehrman sees conversion as primarily being driven by small-scale interpersonal word of mouth: people converting friends, family and colleagues.
Paul is an outlier but Ehrman doesn't see his method as primarily being the 'stand up in front of huge crowds model we think of today, but rather working alongside people and talking to them about his vision of Christ etc. while doing so.
I think in terms of success we also have to ask questions about retention as well as getting converts: people do after all join religious groups and then leave. Ehrman dismisses the 'Christian community/charity' as a reason why people joined in the first place (as it's not mentioned in the accounts and may not have been clear to outsiders) but it might be a key factor in why people didn't leave again. Plus the exclusivity of Christianity means that joining would have involved a commitment and maybe burning bridges elsewhere.
I don't entirely agree with the above commenter, and so would like to provide my 2 cents as well.
First of all, it is true that, at the time, there were A LOT of different cults competing for popularity in the Roman empire. Also, early Christianity didn't count for one of these, but for hundreds of them - as we know today, the Christians consisted of a multitude of disparate groups, that believed different things, had acquired different holy texts, and gained followers in different ways. There were also other Messianic groups, like those swearing to John the Baptist or Melchizedek, and Christianity would seem to just be many among many. While research still isn't clear on why exactly our Jewish Messianic Mystery Cult won in that arena, here are a couple important suggestions:
Conversion. While I don't think it's true that Christianity won out because it was so 'simple' and all you had to do was love and don't sin (becoming a Christian was not an easy task, and one spent years as a catechumen before being allowed baptism), Christianity was special in its openness. Basically, Christian groups were very early to develop a system of conversion, whereby learned and charismatic Christians would travel all over the empire and convert groups of people (from all social classes, mind you) to the faith. This seems like something obvious to us today, but niche cults back then really did not work like this - they were usually rather sedentary, or at least very picky about who they were willing to give the elevator pitch. The primary contender for religious power in the Empire at the time, the Mithraic mystery cult, was reserved for warriors, usually of the upper classes, and, as far as I know, simply didn't spend much resources on public conversion. But why was the JC fan club so insistent on the travelling salesman strategy? And what made them different from other Jewish cults, who would also actively convert people at the time? The simple answer is...
PAUL THE APOSTLE! Now, this may be slightly controversial from a theological standpoint, but I believe that Paul is the single most important person in all of Christian history - perhaps even more important than JC himself. See, first of all, Paul was a very effective preacher; his life as a Christian was spent travelling the empire, converting groups of people to the faith. Also, he kept up with the groups that he converted, making sure they stuck to the faith and didn't falter or listen to other, 'false' preachers. This is what most of the genuine Pauline epistles are - letters to his congregations, solving theological issues, scolding or praising them for their activities.
On top of this, Paul was not just a Jewish scholar converted to the new Messianic faith - he was also a learned Roman citizen, able to 'speak the language' of both worlds, so to speak, and to negotiate Greek and Jewish values into a broader Christian package - which leads us to his most important contribution.
Converting the heathens! Paul called himself the Apostle of the Heathens, and according to his epistles, his converts were mostly Greek pagans (that is, Romans and not Jews). This may, again, seem obvious to most today - Christianity is for everyone willing to accept Jesus Christ into their hearts, right? Not at the time, no. As far as we can surmise, until Paul, the followers of Christ were most likely all jews, and his cult was seen as a Jewish messianic cult - and was therefore obviously reserved for Jews. But with Paul, Christianity began its journey from niche messianic cult to worldwide powerhouse, by taking a sharp turn away from its Jewish roots.
There are other reasons as well, obviously, and the above is just a run-down on what I see as some of the most important ones. The reason I haven't commented much on the state of the Roman state religion at the time is that I more or less agree with /u/Syria_War.
There are good answers here but one area currently overlooked is the social pull that early Christianity had. In terms of growth there are no clear demographic figures available but the population of the Roman Empire was around 60 million people. By the time of Constantine it has been estimated that around half the empire was Christian and that it was fully converted not too long afterwards.
The book of Acts (2:44) claims that the believers held all things in common. It's a bold socialist-style statement that may have some truth. The last pagan emperor, Julian (360-363) aimed to return the empire to its Greco-Roman religious roots but writes to complain that the churches are stealing away the common folk as they provide welfare not only for their fellow Christians, but all people in need. In a society with very little welfare (fertility was prised in part because children were a necessary way of supporting you in old age), a group that provided for the physical needs of its members would be very appealing.
Another point to make is that Christianity's appeal to the poor may have been overstressed. Gerd Thiessen for example analysed Paul's letter to the Corinthians and determined that the Greek converts were more likely to be 'middle class' (insofar as that term is applicable to the ancient world). Analysis of the persecution in the Gallic city of Lugdunum also suggests the Christians there were mostly Greek merchants rather than locals. Indeed, Irenaeus, who becomes bishop of the city around the 180s is himself from Asia Minor.
Again, looking at Acts there are examples of random Geeks who attend synagogues and are clearly interested in learning about new religions. One of the comments above referred to the plurality of the Greek religio-cultural context and so it's not as clear cut as a static populace being wedding to a treasured and venerable Hellenic religion.
This is all to stress that it wasn't just theological appeal that was at work but that socioeconomic factors were present too.
I've looked through the answers given and there does seem something left out, so I'll add it. Christianity included a message of how not to be a slave, even if you were.
Rome was a empire of staggering inequalities, not only of wealth, but also of freedom. Around the time of Christianity's initial spread, 35-40% of the population of Italy was enslaved, and the poor had little recourse. The Roman empire was also very male dominated, leaving few freedoms to women. In this context, as other commenters have pointed out, it's said that Christianity spread the most rapidly amongst the poor, slaves, and women, and there are reasons for this, which I'm going to outline below in too much detail.
First, Christianity is often seen as something very different today than how it was seen at that time. Today we often view religion as a collection of beliefs, while throughout history religions have often been more about being a collection of stories (many of which have teaching). People weren't going throughout the Roman empire giving out copies of the Apostle's creed, they were going around sharing a collection of stories with people who learnt them, shared them, and lived by them.
Second, early Christianity was not a "new Testament" thing, it was very much an old testament thing as well. The early Christians were Jews, they preached to Jews, and they told Jewish stories. When they branched out, it was a massive decision that broke social and cultural norms.
Like many cultures, Jewish people had several stories in their arsenal, and I'll just summarize them. There were creation stories (how the world works, the relationship of man to creator), then they had their stories of the beginning of the lineage/nation (Abraham, Isaac and Jacob) then they had their meat and potatoes story (the exodus, wandering in the wilderness, 40 years of learning how not to be slaves), then Joshua and judges (running an anarchist society) followed by a general history of how things went wrong (Samuel through to Kings). Following Babylonian captivity you have a population reduction from something like five million down to 50 thousand with Ezra, and the deliberate rebuilding of a nation/culture from texts. Out of the rest of the books you have Job (is God just?), Ecclesiastes (is there meaning in life?) and Daniel (here's some hope for the future), and the rest are messages to people who are ignoring everything I've just listed, either of rebuke (always to kings and their people) or of hope.
Now whether you agree with the prophetic messages of Daniel or not, there's no question that the early church at least interpreted the 70 week prophecy (490 years) Daniel 9:20-27 as referring to the coming of Christ and the spreading of the gospel outside of Judaism, and time(d) the prophecy as indicating the year Christ began his ministry, 7 years before the end of the whole prophecy. the line "In the middle of the ‘seven’[i] he will put an end to sacrifice and offering" was understood as referring to his death after 3:5 years, and the end of the 7, the end of the prophecy, was believed to be with the stoning of Stephen, which is why that narrative is given so much prominence in the book of Acts. Here's Daniel 9:24, giving the prophecy:
“Seventy ‘sevens’[c] are decreed for your people and your holy city to finish[d] transgression, to put an end to sin, to atone for wickedness, to bring in everlasting righteousness, to seal up vision and prophecy and to anoint the Most Holy Place.
Acts goes into detail on the stoning of Stephen, his trial before Jewish leadership, his defense given, the rejection of this defense, and immediately afterwards, Peter has his vision regarding "not calling unclean what God has called clean" followed by a knock on the door from the representative of a Roman Centurion, asking him to come share the message with him, and he does.
The politics of this are massive, and take up a bunch of what is left of the book of Acts, with debates over what parts of the cultural heritage of the nation is to be shared, how to separate the cultural heritage from the core message to be shared (the debate over circumcision) and it sets up much of the remaining story of Acts, including Paul's arrest and eventual death in Rome.
So now in this context, the New Testament is made up of the following - 4 books telling the story and teachings of Jesus (including his interpretation of the old testament), plus the book of acts, which is predominately focused on this shift towards sharing the cultural heritage the church believed Jesus embodied with gentiles, followed by a whole bunch of letters interpreting Jesus in the context of the old testament, capped by another prophetic book giving some context going forward. There's some 850 quotations from the OT in the NT, with many more references as well.
At least at the beginning, early Christians went out then and shared these stories - preaching the stories of Christ about non-violence, obedience without being broken and so on (pax Christi in Pax Romana) along with stories of the exodus, Jewish concepts of God, creation, as well as the stories of Acts, justifying the sharing of these stories with new people. A lot of the letters as well deal iwth this topic, emphasizing that we are all now children of the promise to Abraham, so our lives all have meaning.
What this means is that christians and christian missionaries went out giving people the following: a promise of future salvation (life after death, at least somewhat appealing), a path to current mental freedom (going the extra mile, coals of fire, being kind, community, and more) as well as a history that gave meaning to people, often people who had no power, no family, no connections, and no hope.
While I'm sure that the "salvation" part was important, given that Christianity grew most rapidly amongst the powerless, for me the evidence suggests that it was the second and third parts of this message that had the greatest impact on the spread of early Christianity.
For sources, I have references the books of Daniel, Ezra, Acts, Hebrews, the gospels, as well as the journal article "Human Mobility in Roman Italy, II: The Slave Population" by Walter Scheidel. Regarding the view of the early Christian Church and the book of Acts regarding taking a historicist interpretation of Daniel and the day/year principle, we know that Josephus used historicist approaches, as well as commenters from the early church such as Victorinus of Pettau.
As to the main message of the book of Acts, here's an outline showing the extent to which its structure is built around the theme of justifying the passing on of Jewish stories to gentiles: 1 - Jesus leaves, 2 Pentacost (message to other Jews) 4 - conflict with Sanhedrin and sharing posessions, 5 - divine punishment for those not sharing, and more persecution from authorities, 6-7 the stoning of Stephen, 8 - serious persecution begins and Phillip preaches to the Ethiopian Eunuch (unclean on two accounts). 9 - Saul's conversion, 10-11 - Peter starts preaching to gentiles and discussion, 12-14 -preaching journeys to Jews and Gentiles, 15- a council dealing with questions of gentiles, 16-21 more preaching, 22 - Paul returns and is arrested following controversy and accusations, then Paul goes to Rome, the book concluding with Acts 28:28-31:
28 “Therefore I want you to know that God’s salvation has been sent to the Gentiles, and they will listen!” [29] [b]
30 For two whole years Paul stayed there in his own rented house and welcomed all who came to see him. 31 He proclaimed the kingdom of God and taught about the Lord Jesus Christ—with all boldness and without hindrance!