World War I challenged the legitimacy of existing institutions and elites because it undermined their claim to be able to act for the good of the people.
This war came at a time of great faith in human progress. It was felt that human society was inexorably improving. The institutions and elites that ran society, therefore, could be seen as good because they were part of that progress. People could trust their institutions and elites.
But then the war came. The institutions and elites were not able to prevent the war. Worse, they showed that they were willing to destroy millions of lives and huge amounts of wealth fighting the war. All of a sudden their leadership did not seem so beneficial and their legitimacy was called into question.