Rome had, since the end of the Severan dynasty, suffered from tremendous internal fracturing. In-fighting and strife among emperors and generals who contended for authority ran rampant. Additionally, the incursions by the barbarians created something of a "revolving door" effect in the imperial lineage. Because news traveled (by modern terms) slowly, there were sometimes three or four emperors ruling at the same time. When Diocletian seized the throne, he codified this, by establishing the tetrarchy. Now you are probably wondering where the monocentrism of Christianity might fit into this. Constantine, desiring sole rulership, used Christianity as a way uniting the empire: One God, one empire, one emperor. One of the important things to remember about Christianity is that division is a terrible, terrible thing. The apostle Paul, in literally every letter he wrote to a church, strongly encouraged them to be unified. Jesus did the same thing. For an emperor hoping to use religion as a sort of "glue", this was very important. Christianity, however, constantly had to deal with heresy, literally from the beginning on. Disputes over heresy are all over the Bible (just look at the epistles of John, or the prophecy regarding the Nicholaitans in the book of Revelation), and they continued throughtout early church history. Constantine had to fix this problem, so he did something that the church hadn't done in 300 years, he called all the bishops together to make a decision, and ensured that they did come up with one. This was the council of Nicaea, the first in a long line of church councils. Now what made accepting this new found religion so easy? Several things. The Roman world already had a monotheistic deity known as Sol Invictus, and making the transition from that to Christianity was not a terribly large leap. Even after Constantine's conversion he continued to portray himself as a sun deity, sporting the traditional solar crown. Additionally, the imperial "job description" had changed since the days of Augustus. Ammianus Marcellinus, a late Roman historian, shows very clearly that monarchy was not something the Romans had to hide anymore, as in the days of Augustus. Even in Tacitus' day the farce was over. By the time of Constantine, however, the office of the emperor had taken on godlike importance. Since the time of Augustus, there had always been an imperial cult, however the general "theology" if you will changed as well. By the time of Diocletian, emperors were living deities, relying on very eastern models of kingship. Even the imperial dress began to resemble something more along the lines of a chinese emperor. Suffice it to say, many things had changed since the time of Augustus. From a more utilitarian standpoint, Constantine very publically supported Christianity. Anyone with a brain knew that if they wanted to have a political career, they would have to convert, which they did. With the elites of the society converting, the ordinary people followed suit. Christianity was still not the official religion, however. Pagans were still free to worship their gods as well. It was not until emperor Theodosius that this really began to change. Theodosius made Christianity the official state religion, however paganism continued well into the 6th century in the eastern empire. Some additional sources to look at:

Ammianus Marcellinus, The Surviving Books of the Histories of Ammianus Marcellinus. With English Translation by John C. Rolfe. Cambridge: Hardvard University Press, 1963.

Timothy D. Barnes, Constantine and Eusebius pg. 43. Cambridege: Harvard University Press, 1981.

Eusebius. "The Life of Constantine." in Nicene and Post-Nicene Fathers, 2nd Series. Vol. I. Edited by H. Wace and P. Schaff. Translated by Ernest C. Richardson. October 10, 2013.

Arther Ferrill, The Fall of the Roman Empire: The Military Explanation. London: Thames and Hudson Ltd., 1986.

I'll list some of the good works on this topic below. For now I'll sum up some of the arguments. MacMullen assumes these mass conversions, starting in year 40 and ending with 5-7 million Christians by the year 300. His primary reasoning behind this is that the Christians performed miracles. My critique here (all learned from Bart during a seminar with him) is that Christianity actually didn't have to mass convert, the numbers show that Christianity actually just had to grow by a very small, steady percent each year (3-4%) to jump from that 20-40ish, to 3-5 million by 300 or 400. Martin Goodman argues that it was because of the Christian evangelical movement. He argues this was a novelty in the ancient world. For the most part, I agree but still it didn't grow at a very fast rate which Goodman assumes. The work that really shows this numbers argument is Rodney Starks, The Rise of Christianity. He shows how Christianity didn't have to grow at huge rates every year. Remember, a convert for Christianity was a loss for pagans. Whereas a non-convert did nothing against Christianity. He then speculates exactly what you asked, why were they prepared to accept the monotheistic faith. in other post (somewhere in my comment history) I wrote about how the Romans did already have ideas that weren't too far from monotheism, such as henotheism and Platonic monotheism. Some philosophical schools actually did teach that there was just one god, and that it was evident in the orderliness of nature. Also, the pagans of the empire already did believe in magic and miracles. So when you had the followers of Jesus and later Christians performing these miracles, they did so with the argument that if this miracle succeeds, then it proves there is only one God, the Christian one. This is what was novel, that if their magic succeeded, then it proved the existence of one God. I know that it may be hard for some to think that people believed in their magic or miracles, but people really did. Even more important is, whether or not it happened, the news spread that these miracles happened. Like I said, many people didn't have to convert every year, just at a constant rate. There's so much more I could type about this but unfortunately I have to go :(

I want to point out that Bart Ehrman has a forthcoming book on this material. I was fortunate enough to read this. He has also led a seminar on the topic which I participated in.

The other books I've mentioned are: Rodney Stark, The Rise of Christianity Martin Goodman's Mission and Conversion Ramsay MacMullen Christianizing the Roman Empire

Edit: Also, a lot of people will say Constantine was instrumental. In some ways he was, and in some he was not. At the rate that Christianity was going, it would have either been him that converted or another emperor soon after. Plus, his conversion was not pivotal and did not change the rate of conversion much. BUT, he was pivotal in allowing Christianity to operate with imperial patronage. We really see anti-pagan laws pick up though in the fourth century and after, not with Constantine. Constantine and his immediate successors were much more concerned with unifying Christians and rooting out heresy, rather than turning the empire Christian.

Edit 2: Trombley's groundbreaking work Hellenic Religion and Christianization does an amazing job of looking at the laws that were issued post Constantine, as well as their effectiveness and enforcement.

Edit 3: Also, James Rives (and others) make the argument that Christianity encapsulated different trends present in the Roman world but did so in one package.