The concept of three incarnations of Rome is connected with and based upon religion, geopolitics, and nationalism.
When the first (the actual) Rome fell in CE 476 it meant that Constantinople, the seat of the Eastern Empire, now was the only "Roman" capital remaining. It was, however, the center of Greek, not Latin culture. Though Rome was still the seat of the Papacy, the fact that the Western Empire had fallen to "barbarian" kings meant that the conjunction of imperial and ecclesiastical rule now existed only in the East, in the city on the Bosporus. Therefore, Constantinople became the second Rome, taking on fully the role of leader of the still unified, at least theoretically, Mediterranean world.
When the Great Schism between Eastern and Western churches occurred in the eleventh century, from the standpoint of Eastern Orthodox Christianity Constantinople was now the only valid seat of the True Church. But the city fell to the Ottoman Turks in 1453 and the great church Hagia Sophia ("Holy Wisdom") was turned into a mosque. The entire world of Christendom was stunned. Though the Turks allowed the Greeks and other Christians to continue practicing their religion, the combination of temporal and spiritual power of which the city had been a symbol for over a thousand years no longer existed as such. But the rising empire to the north, Russia, was there to take its place. Because the Russians had liberated themselves from two centuries of Tatar rule and now had pretensions to imperial power themselves, and because their form of Christianity was the same Orthodox division to which the Greeks adhered, Moscow was now positioned to inherit the mantle of religious leadership, plus temporal power that Rome and Constantinople had symbolized. The center of such power had thus moved from Rome, then east to Constantinople, and then north to Moscow.
The concept of Moscow as a third Rome really had validity only for Russians themselves and for the broader russophile or russocentric world. Western Europeans had believed the new Rome was located in their own lands at least since Charlemagne's ascent in the eighth century. For a thousand years, with shifting boundaries and populations, the Holy Roman Empire continued its existence until Napoleon abolished it in 1806. According to Voltaire, it was "neither holy, nor Roman, nor an Empire," and Gibbon wrote that Vienna was the seat of "a German prince, who styles himself the emperor of the Romans." These statements attest to the fact that (in the view of freethinkers, such as Voltaire and Gibbon) the attempts to reestablish "Rome" were a kind of recreation of a mythic concept, rather than one based in concrete reality. But both the Holy Roman Empire and the status of Moscow as the third Rome showed how powerful the concept of Rome, now in the guise of a resurrected power center, was for all of Europe, both west and east. Even now the EU is an attempt, at least in the form of economic reality, to unify Europe on the territory of the former Empire.
In Tolstoy's War and Peace, before Moscow falls to Napoleon, an inmate of a mental asylum seems to conceptualize himself as a symbol of these successive incarnations of Rome when he cries, "thrice have I fallen and thrice will rise again." This is a prophecy uttered by a mentally disturbed man, but it is also an expression of the Russian nationalism that will defeat Napoleon. It's emblematic of the manner in which Russians saw their Motherland as the embodiment of both Christianity and power.