How did religion influence the development of europe during the middle ages?

What happened, was that Europe was converted from Pagan (Roman), to Roman Catholic. 

After the fall of the political state of Rome, in the West, (the Empire continued in the East), the strongest social, and educational institution remaining was the Catholic Church, under the various Popes. 

The Catholic Church had it's schisms, spiltting too, and at one time there were at least two Popes, one of them in France. 

But, eventually, because the church had it's central bases, and main power in the larger cities, it grew in power. Merchant classes supported the Church, due to the Church's spending. It built huger cathedrals, and provided hospital care for the ill, and food for the hungry. 

Eventually the countryside converted to Catholicism, because they had to deal with the cities, and merchant class. They also required the social services of the central church. 

Because most "heathen", pagan peoples believe in multiple Gods, it was not difficult for them to "convert" to accept another God, Jesus, or priesthood, because they were polytheists. 

Eventually, the Church was strong enough to persecute and root out those with different dogma. Blasphemy became a capital crime. 

But, overall, the Church (not just religion) helped unify Europe into many different strong states. 
For a while, the Pope had to "bless" or recognize the Kings, or main leaders too. Without the blessiogs of the Pope, a leader was doomed.

There is lots of evidence