The major geopolitical circumstance that led to the increase in the membership of the United Nations in the 1960s was decolonization. This decolonization was brought on in large part by World War II and by the Cold War.
Before WWII, Britain and France had held many colonies, largely in Africa. The war helped to reduce their ability to hold an empire. The war also led to the Cold War, which made colonization less possible. The countries of the West had to liberate their colonies to help prevent the communists from being able to portray them as imperialists. Because of these factors, the British and the French, in particular, granted independence to many colonies. The growth in United Nations membership in the 1960s came almost exclusively because these new countries were entering the organization.