Islam, for example, was not welcomed by the Arabs, and over time Mohammed built an army and eventually made non believers surrender to it. Similarly, it spread throughout the world, using force, wars, and violence. I have simplified things a lot here, but more or less we know accurately how Islam dominated Hijaz and then other parts of the world.
However, Jesus never built an army or a massive number of followers while he was alive (if I am not wrong). Then — due to my lack of knowledge — it seems that all of a sudden it becomes more and more popular until it virtually obliterates all other competing religions of its time (in Europe).
Do we know how exactly it started to grow and why people, if I am not wrong, willingly converted to it?
Quoting myself from a previous answer to a similar question:
The growth and spread of a religion is an incredibly complicated thing but Christianity had something going for it that was hugely influential in its eventual primacy in Europe: accessibility.
Christianity, especially as it was preached by Paul and his followers, was incredibly accessible to people of low social rank. Unlike many mystery cults that had fees to join, Christianity was free to join and actually encouraged its members to support each other and to support those in need. This allowed it to spread rapidly amongst the lower classes; even if someone originally came to their meetings for the free food, eventually they might come to believe the message. Beyond that, it had the very enticing message that there was a better life after the current one. For slaves and servants this would be incredibly attractive.
It also had all the allure of other 'foreign' religions (like the cult of Isis or Mithras or even Judaism itself) but none of the drawbacks (fees, difficult to follow rules, edicts of cleanliness). Paul preached that gentiles (non-Jews) did not need to abide by the Jewish Laws. This just makes becoming a Christian that much easier for someone who is interested.
Over time, Christianity spread. First mainly among the lower classes, but then among the upper classes in growing numbers. By 300 AD, as many as 10% of the people in the Roman Empire were Christian.
In 312 AD, Constantine (who was fighting for control of the empire), for reasons that have been debated for centuries, decided to accept Christianity. The shift wasn't instantaneous. In fact, we don't know much about Constantine's Christianity (he wasn't baptized until the end of his life and there is no mention of it on his triumphal arch) but he did make religious tolerance a cornerstone of the empire. This tolerance was all Christianity needed to spread like wildfire.
In 380 AD, the Christian Emperor Theodosius' Edict of Thessalonica made Christianity the state religion and ordered all subjects to participate. This was effectively the end of paganism in Rome.
You might be interested in this older thread:
Do we know that much about the rise of Islam? I certainly don't think we can say that we know exactly how it arose and about its motivations. The same goes for Christianity - I really don't think we can pick apart the complex mind of an individual Roman and say exactly why he or she converted. We can note general trends and show that it happened, but uncovering motivation is I think very difficult. I attempted to do so in this comment, but all it did was make me think more about this process. There were I think many reasons for people to convert, which was sometimes hindered or aided by the political and cultural climate. Other than that, I think a healthy degree of doubt and thinking outside the box is warranted for this topic (and frankly the same goes for any historical process).