On the one hand, the Cold War had never been confined to Europe. The early years of the conflict saw struggles in Korea, Iran, and in Taiwan. But it was in the 1960s and 1970s that the Cold War entered the "Third World," itself a term of Cold War construction. To some extent, this was a result of the hardening, and mutual acceptance, of Cold War boundaries in Europe. Neither side was willing to risk global conflict over Berlin, for example, and so the struggle shifted elsewhere.
Perhaps the main reason the conflict shifted to the Third World, however, was decolonization. After the war, European colonies in Africa and Asia began to assert their independence. When the colonial powers resisted independence with violence, as in French Indochina, the United States and Soviet Union chose sides in the ensuing conflict. Sometimes this took the form of "proxy wars" in which Even when violent conflict did not ensue, both powers sought to influence the "state-building" process, as in India and Egypt.
The US and Soviet Union even tried to intervene, often violently, with hostile regimes in established states, as American actions in South America, especially Chile, demonstrated. Often these actions were motivated as much by threats to Western business interests, as in Iran and Cuba, as by ideology. But the Soviet Union, in particular, was successful in making ideological appeals to nationalist leaders who pushed for land reform and more equitable allocation of resources. While there is not space here to discuss all of the examples of superpower intervention, the expansion of the Cold War took place in the context of the stabilization of the conflict in Europe and the collapse of European colonial empires as well as nationalist movements in Latin America.