A number of factors lead to the rise of the Roman church as the preeminent authority in Western Christendom. The following answer is painfully over-generalized:

You need to start with the context, which involved the dissolution of the Western Roman Empire, and the manner in which the church became enmeshed in filling the power-vacuum that followed. During the late Roman Empire (300s and 400s) the Trinitarians (like the Roman church) were in conflict largely with the Arian faction of Christianity, which lead to networks of bishops linked to one group or the other throughout the empire and beyond.

The ability of the Bishop of Rome to bring other bishoprics under its control during this time- using influence, wealth, appeals to theological and charismatic authority and other forms of soft power- put it at the center of a network of churches that gave it authority throughout Western Europe despite political and social transitions in the early Middle Ages, pushing out the Arians.

Working together with local and national authorities, the church was then able to use political power to treat dissenters and rivals as threats, legitimizing their alienation and persecution in the name of holiness and/or social order. A strong network of monastic institutions provided further influence and served as educational institutions which further spread their form of Christian belief.

By the 1100s the ways this partnership with secular powers had shaped the church as a powerful, arguably too-worldly institution was evident enough to lead to the reform movements which would eventually lead to the Reformation, which started out as an attempt to change, not leave, the Western church.

Ultimately the rise of the Western church had mixed results: it provided a common system of belief among various nationalities, which provided both stability and the basis of an idea of 'Europe', and also created the first universities, but it also stifled dissent and allowed itself to be influenced by the politics of the day to a very large extent.

Edit: the centrality of bishops to this answer needs clarification. Bishops held immense influence in the church from the 3rd century on for a number of reasons. Three I want to mention are:

1. The theory of Apostolic Succession made them the successors of Jesus' disciples, which granted legitimacy. The Bishop of Rome claimed spiritual descent from Peter and Paul, and his blessing on another bishop granted even more legitimacy.
2. Bishops performed many of the sacramental duties that priests would later acquire. For centuries only a bishop would baptize, literally giving them salvific power.
3. By the Middle Ages Bishops controlled as much property as many nobles, and in some cases were treated as nobles, giving them immense secular authority as well.