There are at least two different ways in which to answer this question.
On the one hand, we might take the question to mean something like “how did the Third World become more developed (or advanced) during the time period 1940 to 1975?” If this is the question, the answer is that the Third World did not really become very much more developed at all. That is why it remained as the Third World and it is why most of these countries remain poor today. We can say that much of the Third World developed in that many of these countries started this period as colonies of European powers, only to become independent. We can also say that some countries that started the time period in the Third World moved up and reached (or at least approached) the First World. An example of this would be South Korea. Such development as occurred came about largely because of globalization. Third World countries benefited to some degree from aid from the outside and from trade with richer nations.
On the other hand, we might take this question to mean something like “how did there come to be a Third World during the time period 1940 to 1975?” In this case, the answer would revolve around the Cold War. At the end of the 1940s, the world began to divide up into blocs that were allied with the United States or the Soviet Union. Most of the rich countries of the world became part of the US bloc and were called the First World. The countries that became communist were generally seen as the Second World. The term “Third World” was used for countries that were very poor and were not closely aligned with either bloc. These neat categories were somewhat hard to maintain, though, since countries that were clearly poor, such as Cuba, became part of one bloc or the other. However, we can say that it was the Cold War that caused us to develop the concepts of the First, Second, and Third Worlds.