Well, the Romans did actually often persecute Christians (and others, mind you). But Christianity grew over time as a mostly underground sect within the Empire.

When the Emperor Constantine finally adopted Christianity, that was when you basically got the start of "Roman Catholicism".

http://en.wikipedia.org/wiki/Emperor_Constantine

You can read about it at the wikipedia article which is actually pretty good and has lots of sources and a fairly decent synopsis.

Also see the entry about Constantine and Christanity:

http://en.wikipedia.org/wiki/Constantine_I_and_Christianity

Notice that Christianity wasn't legalized until 313 AD by the Emperor Constantine, which means that Christians went for hundreds of years without any protective rights for their religious views.

As to why Christianity survived for so long without any legal rights in the empire (three hundred years, after all, is a very long time), there are many theories. Many Christian converts were apparently slaves. There were also those who were seeking more "exotic" religions to delve in, and Christianity was one such religion on offer. That's an entire field of study in of itself, however. But yes, the Romans often looked down on Christians and saw them as fundamentalists, "crazies", and even as criminals who wouldn't worship the Emperor or pay tribute to Roman gods. In fact, it's the same reason they persecuted Jews. Jews wouldn't pay tribute to the Emperor or the Roman gods, and were seen as fundamentalist barbarians by the Romans.

The fact is that the Romans generally didn't care what kind of religion you had, as long as you paid tribute to the Emperor and their gods, you could pretty much do what you wanted with your own religion. Many religions adopted Roman customs over time and were basically left alone. But Christians and Jews simply refused to do this because of their monotheism (which was a fairly new concept to most of the people in the Empire at the time).