The author means by the phrase that People were able to choose who would lead their governments.
Explanation:
In the 20th century there were a lot of new nations that were formed out of old imperialistic colonies and most of them adopted, or ended up adopting a democratic system of governance where the people have the right to choose who would govern over them and the positions are rotating.
This is something that could be called to have given greater ability for the people to govern their own lives and the author has teemed the century thus as the century where the [power went to the people.