Why was it Christianity and not Judaism or Zoroastrism?
The differences between Christianity and Judiasm in this regard were 1) Christianity wanted to spread 2) Judaism had high barriers to entry.
The early 1st century AD in the Eastern Roman Empire was a time of religious and cultural upheaval. Traditional Greco-Roman paganisms were losing their religious vigor, and becoming gradually more cultural and national practices than sincerely held religious faiths. Second Temple Judaism was divided between various sects, and significant proportions of Jewish people were assimilating to Greek culture (Hellenized Jews), meanwhile many non-Jewish residents of the Roman Middle East found Judaism attractive and basically believed in it but did not take the final steps of joining, this latter group was called the "god-fearers".
Judaism at that time (and in the modern era) was not really a religion that sought converts. The Jewish faith required people to keep kosher and required men to be circumcised, and adult circumcision is not an attractive option for most people without modern medical care.
Christianity had its roots in Judaism but the Apostles quickly determined that they should be sharing their faith with Gentiles. This is recounted in the story of Pentacost in Acts of the Apostles where the Holy Spirit lets the Apostles speak all sorts of different languages so they can preach to everybody.
St. Paul, an early persecutor of Christians, also converted to Christianity and became determined to spread the Christian message to the gentiles. He spent a lot of time traveling around the Eastern Roman Empire converting both Jews and Gentiles, particularly the god-fearers, who were low-hanging fruit.
The most significant controversy of the Early Church was between James the Just, the Bishop of Jerusalem, who believed that in order for gentiles to convert to Christianity they also had to become Jewish, so keep kosher and get circumcised, and St. Paul, who believed that gentile Christians should not have to do any of those things. St. Paul won that argument.
In 66 AD, the Jewish Revolt broke out in the Middle East, and as a consequence of the Romans crushing the revolt, the Second Temple was destroyed, and with it Second Temple Judaism and the various sects. This was the real point of division between Judaism and Christianity. While Jewish leaders regrouped and laid the groundwork for modern, rabbinical Judaism, Christians continued to expand across the Eastern part of the Roman Empire. They quickly converted most of the god-fearer communities and a significant proportion of the Hellenized Jews.