To build upon /u/ddsilver's answer, I'm going to give you the theological rationale that allowed the early Christian church to become not-exclusively Jewish. i.e. the historical shift to the inclusion of Gentiles is undergirded by a theological shift bound up in the earliest christian documents.
If you want two primary documents to read, read the Acts of the Apostles, and (Paul's) Letter to the Galatians. Acts is giving you a primary source dealing with how the followers of Jesus spread from being Jews only, to including Gentiles. In fact, arguably the book is structured to show this. Acts 1:8 sets up a geographical paradigm (Jerusalem, Judea, Samaria, ends of the earth) which corresponds to major shifts in the narrative.
Key moments occur in the narrative of Acts when Peter the apostle has a vision in Acts 10 that Gentiles are acceptable to God; he then goes and shares with non-Jews, and arguably the presence of the Holy Spirit on these Gentiles is a sign that they have already been accepted by God.
Acts 15 is the other major turning point to look at. The so-called Jerusalem Council meets to consider whether believers from Gentile backgrounds need to become Jews or keep aspects of the Mosaic law. The answer is minimalist.
Galatians, if it is Pauline, shows Paul combatting a group that things following the Jewish Messiah must first mean becoming a Jew. Paul's argument is, in a nutshell, that the promises set up to Abraham in the OT overarch and supercede the Mosaic covenant, so that faith in Jesus is the key marker, not being a ethno-religious Jew.
Essentially early Christianity proclaimed a Jewish messiah as the fulfilment of the Jewish religion, but interpreted that fulfilment in a way that meant Gentiles could become followers of the messiah without first becoming Jews. Because, from their perspective, Jesus isn't only 'King of the Jews', he is King of everything
The short answer, which is more theological than historical, is that Jesus dissolved the covenant between the Jews and Jehovah, effectively invalidating the Jewish faith. It was replaced with the standard tenet of Christianity - believe in Jesus and be saved.
Take a look at this list of bible verses: http://www.openbible.info/topics/new_covenant. That covers the establishment and significance of the New Covenant pretty well.
Clearly, this was unpopular with the Jewish leaders at the time, and they took some steps to "deal" with Jesus. In a manner of speaking, the actions of the Pharisees preserved the Jewish faith to the present day.