This happened in a number of ways.
First, the war revealed to Africans the fallacious nature of the idea that the Europeans were superior to the Africans. Any continent that could let itself get devastated by two huge wars in thirty years could hardly be seen as morally superior.
Second, the war led to many promises to Africans. The British and French needed the help of their African colonies to win the war. They needed the people to work willingly to do things like extracting resources that would be needed for the war. To get cooperation, they made promises about increased rights for Africans. This created a sense of heightened expectations after the war.
Finally, the war led to an international system led by the United States. The US disapproved of colonialism and was not typically willing to support its allies' desire to have colonies. This put more pressure on the French and British to give up their colonies and gave the Africans more of a sense that their time had come.