Well, in general, I think we can safely say that your question is simply too broad to properly answer. "Medieval Europe" spans a thousand years and even more miles. I would recommend you to divvy up the subject, so to say, and treat it as several different questions.

In general, if you like a subject is too broad it probably is and I don't think there is any good way to summarize medieval European religious life.

I do think, however, that we can say something about the relationship between Christianity and the crusades. Namely, Christianity and the crusades and inseparably linked, but there were other factors beyond piety that made them such a large enterprise.

First and foremost, what we call the first crusade, preached by Pope Urban II was actually not the first crusade called by a pope. In the 1070s Pope Gregory VII sends out many letters calling asking for militarily inclined men to assist the Byzantine Empire again the Seljuk Turks. Nobody really responded to his call.

In 1095, Urban II makes a similar plea with a few slight differences. One, he does not send a letter but instead announces the crusade at the council of Clermont in person. Second, Urban singles out the capture of Jerusalem as the specified goal. Third, according to all 3 records of his speech, he promises all how take part in the project automatic entry into heaven should they die. This was quite an attractive proposition, as even the most grievous sinners could alter fate in the afterlife by embarking on this journey. Add to this that crusaders, both pilgrims and knights, stitched crosses onto their clothing to symbolize their missing and chanted Deus Vult, latin for 'god wills it', it becomes difficult to separate Christianity from the crusading enterprise.

People from all walks of life answered Urban’s call. In fact, an army accompanied by a very large number of peasants left for Jerusalem before the appointed time. According to Ekkard of Aurach, people from “...Aquitaine and Normandy, England, Scotland, Ireland, Brittany, Galicia, Gascony, France, Flanders, Lorraine, and from other Christian peoples, whose names I no longer retain” took up the call to arms.

However, there was much more at work than sheer piety. Remember, Gregory VII made a similar call and received nowhere near such a large response, so clearly the pope calling for a crusade was not sufficient to elicit such a response. As such, we must turn to other factors to explain why people went on crusade.

First, part of the popularity of the crusade stems from elite military culture itself. As part of a knight's training he was expected to travel around with a group of other young knights and do knightly things, namely fight and cause trouble. France was full of men like this. They traveled around looking for fights to attach themselves too and when no fights were to be had started their own. These little wars got so bad that many churchmen tried to institute 'peace days' where fighting was forbidden on say every odd Tuesday and Friday.

When Urban calls a crusade, he creates an outlet for many of these young fighting men. instead of fighting each other at home, they now have a chance to travel abroad and fight for glory and god. This was an especially attractive proposition for many, as it provided an opportunity for young elites to not only exercise their martial valor, but to do so on God's behalf. Some historians argue that the crusade was a deliberate attempt to flush these men out of europe to curtail the violence of the time. I think that this is a stretch too far, but it would not be unlikely that Urban knew of the advantages of reducing the number of fighting men in the heart of France.

TL:DR. There were additional factors that made the crusades, especially the first one, possible, but Christianity is both central and inseparable from the crusading mission.