Greetings in that Divine and Most Precious Name of Our Lord and Savior Jesus Christ!
Perhaps this speaks to my Protestant hang-ups, but what kind of "bridge" are they building? Does Christ the one Mediator between man and God need someone, say, to mediate between us laity and Himself?
As I think the theological aspects were explained already I'll tackle the historical. The pre-Christian Roman Pontifex maximus was a title used through out the first to fourth centuries it was reserved that that the Roman Emperors would have what me might say, multi-jurisdictional authority over the various Mystery cults of the Empire, including the first century Jewish Temple clergy. The Roman Emperors serving as this title were intended to function as a kind of mediator for disputes and also a unifying "bridge" between the differing religious institutions. I think then that in the context of the Church, the Pope is intended to be the maximum bridge between the differing parishes and regions, to really implement a Roman Catholic concept of the Universal
This historical context of the Roman Emperors is a good way to help understand some factors which played into the increasing centralization of the authority of the Popes in Rome. At different times when Popes were going through what we might be able to think of as a "throw back" era relying on proto-nationalistic revivals of the old Roman glory, they also fell back on the universality of the Roman Pontifex maximus. When the Popes in Rome began to assert a dominant role in the worldwide Church, they were relying on their own historic sense of "being the big brother" (and in the school yard sense, not the 1984 sense) for the world as the Roman Emperors had been before Christianity. As the Roman Imperial Pontifex Maximus was the head and mediator amongst the disputes of the religions of that Empire, so to did at time the Popes feel somewhat historically entitled to the same privileges within the One, Holy, Universal, Apostolic Church. Of course the other jurisdictions have always disagreed. This of course may also be the influence of the Caesaropapism of the Byzantine Emperors, who were equally influenced by old Roman, pre-Christian Pontifex maximus of the Roman Emperors.
The Byzantine Emperors since Justinian were essentially and functionally the unbroken perpetuation of the Roman Empire, so just as the pre-Christian Roman Emperors had served as the Pontifex Maximus of the Empire, so to did the Byzantine Emperors as their successors claim the same authority. This is what has been called "Caesaropapism" in the context of the Church, since the Emperors functioned as secular heads of the Church at different times and over different jurisdictions of Orthodox. The Emperors realistically sought to achieve the same kinds of universal, old Roman authority of the religious institutions in the same way the Popes in Rome were. The Popes in Rome only began to claim this title formally after the Schism when they could be free of the Byzantine Emperors sway and international influence (think the UK of its time, not as hegemonic as say the USA but a force to be reckoned with just the same) but really even since Leo III within the internal Roman jurisdiction the Popes maintained their old Roman privileges of Universal authority, something very foreign to the Patriarchs of the East and Orient.
I really like the Pontifex maximus topic, it is a very pivotal concept from the old Roman era and the first century in particular, but its influence really can still be felt even today, aside from some major events in the past 2000 years!