There are two ways in which we can look at the causes of this movement.
First, we can look at the deep historical roots of the movement. When it was first colonized, the area that is now Quebec was French. The French colonized that area beginning in the 1530s. Much later, Great Britain took control of all of the French possessions in North America. This happened at the end of the Seven Years’ War (known in the US as the French and Indian War) in 1763. At this point, Britain was ruling what is now Canada, but the great majority of the people living in Quebec were of French descent. The British government allowed them a great deal of autonomy. This led to a situation where the French in Quebec were able to see themselves as something of a separate nation.
Second, we can look at the more recent causes of the movement. In the mid-1900s, Canada was industrializing and becoming richer. As Quebec changed from a more rural province, the French in Quebec came to feel that they were being treated as second class citizens by the national government. They did not tend to identify as much with Canada as they did with Quebec anyway and now they had a reason to feel that it would be worthwhile to separate.
For these reasons, a movement arose to push for an independent Quebec.