The church is the only institution to survive after the fall of rome, why?

One obvious answer is the Eastern Roman Empire. That evolved over the few next hundred years into the Byzantine Empire. The Byzantine Empire only came to an end when Constantinople was captured by the Turks in 1453. Western Europeans met the Byzantine Greeks when in 1097, they marched on Crusade through Byzantine territory to try to reach the Holy Land. 

In religion, the West avoided the Christological controversies that embroiled the Byzantines to the East. There was an ecumenical council at Nicea in AD 323 that decided that Jesus was both God and human, the doctrine of Athanasius. 

As Western Europe evolved into the Middle Ages, an language division emerged along the Rhine River. To the North, the language evolved into German. To the South, it evolved into French. 

Among all the peoples who invaded the Roman Empire, all of them had been converted to Christianity except the Franks and the Huns. The trouble was that all except the Franks and the Huns were converted to Arian Christianity. The Roman population were what we would call Catholics. The Franks and the Huns were Pagan. 

The conquering Aryan barbarians regarded the Romans as heretics. In AD 511 Clovis, King of the Franks, converted with all his people to Catholic Christianity. That sealed the triumph of Catholic over Aryan Christianity in Europe Hope this helps!!