Why might a country's leaders have wanted to join the nonaligned movement during the Cold War?

Answer: They wanted to avoid the strife and militarization that characterized the relationship between the Cold War powers (the USA and the USSR).

Details:

As the superpowers in the Cold War, the USA and the USSR, sought to line up countries in allegiance with their positions, a group of nations emphasized the importance for remaining non-aligned. One of the leaders of the non-alignment movement, Jawaharlal Nehru, said in a speech in 1948: "When we say our policy is one of non-alignment, obviously we mean non-alignment with military blocs." The Non-Aligned Movement held its first conference in Belgrade, Yugoslavia in 1961. The members of the movement saw the siding up being done in the Cold War as a path to increasing world tension and conflict, and sought to remain non-aligned for the sake of preserving peace and equal opportunity for development.