In early medieval Europe, there was a need for a way to create social and political order. Such order had existed during the time of the Roman Empire, but when that empire collapsed, Europe became politically fragmented. Before Rome, Europeans had existed as members of tribes in small, fairly informal societies. Therefore, they had no set system for how to keep order in a large society without the Roman government. Feudalism was the answer that they eventually developed to this problem.
The bond between lord and vassal was a way to create political order. Without the Roman government and without feudalism, government would have been chaotic. A man could call himself a king, but he would have to try to impose order by force. He would not have anyone who was necessarily loyal to him. This would make for political disorganization. Feudalism solved this problem to a great degree. The oaths of loyalty between lord and vassal gave people a reason to cooperate with one another and to be loyal to one another. This created a framework that allowed political order to be maintained.
In terms of social order, imagine trying to keep order with no police force. Imagine trying to do so without much in the way of roads. Imagine trying to keep order when it was very hard to communicate with places that were more than a few miles away. Effective governance would have been impossible, leaving society to (potentially) fall into chaos. The feudal system helped to prevent this. It created a system where there were chains of authority that led from the top of society to the bottom. It set up local lords whose job it was to keep order in their areas. These local lords had courts for deciding legal disputes. They acted as local governments. By their relationships with their lords, they connected their local area to a broader social system.
Thus, the lord-vassal relationship allowed for political and social order in a society that was technologically primitive and which was still trying to find a new way of keeping order after the fall of Rome.