How did the Crusades help weaken feudalism?
The Crusades in all ways was a Holy War. Many Kings from different lands (Louis VIII of France, Conrad III of Germany for example) led groups of soldiers to take back the city after Turks and other Muslim/Islamic groups would slaughter the Christian pilgrims in the city and burn their churches. On one side, this was a major slight on the Faith of much of the Western world and they felt they needed to defend their religion from the invaders. On the other hand, Jerusalem lays in the middle of a vast desert with very little resources to speak of.
The Third Crusade marked the entrance of the English under King Richard who led his warriors to the city to try to reclaim it. One of the biggest issues with this was that armies are not cheap and England was already not a very rich kingdom at this point. In order to run an army not only do the people need armor, weapons, food, etc. they also need to be paid. This led to extreme taxation of the lower classes under the feudal Lords who required money to ride to war or send their knights.
On top of just getting to the area, King Richard never actually took the city. He attempted to return back home by ship but was abducted by the Duke of Austria. The Duke, whom Richard had slighted during the war, forced the English to pay a ransom worth twice of England's annual revenue at this time.
After all of these pay-outs (which was not just in England, with similar losses in other countries in the West) the peasant and working classes were left unsupported and broke. Lords no longer could keep their peasants under their control as all money taken by the lords was essentially wasted during the crusades. This led to peasants being able to buy land for themselves very cheaply and being able to be their own masters, thereby ending the feudal system.
The Crusades also opened a world unknown to most crusaders to that point. In many ways, the Middle East at that time was more developed than Europe. Medicine and architecture, for instance, were well ahead of those of their European counterparts.
For example, in early Norman times, fortifications used angled rather than circular surfaces. Corners were much weaker than curved surfaces, and square towers could be taken in a number of ways. Corners could be breached, or towers could be weakened to the point of failure by digging underneath the corner.
Circular structures, however, were much stronger to direct attacks, and could not be weakened significantly simply by digging. This circular building technology was already in place in the Middle East at the time of the Crusades, and was brought back to England with the Crusaders. We can see evidence of this today in England, when we look at building dates of the many Norman castles still standing.
In medicine, medieval doctors in the Middle East based their practices on the ancient Greek texts that were available to them, whereas in medieval Europe, medicine was dominated by the tenets of the church. Middle Eastern doctors performed surgeries, used a variety of drugs for treatments and also encouraged and supported the education of women as physicians. As with architecture, the ideas encountered in the Middle East eventually made their way back to Europe with the returning Crusaders. Having witnessed "real" medicine in action, it is very likely that the Crusaders beliefs in the all powerful church were weakened as a result.
The Crusades were the beginning of the end for Feudalism. Many peasants went on the "adventure" as it promised free entrance into heaven, and it promised a life free from the toil and serfdom that they had been born into.
As the peasants made their way on the journey they began to experience aspects of life once closed off to them, including increased pay. Once they returned home they felt they deserved higher pay.
The final nail in the coffin for Feudalism was the Plague though. People quickly began to realize that Kings had no power to save them from the plague, and with the reduced work force left behind they could demand higher wagers (the laws of supply and demand are always at work).