Like many nuclear states, India's route to the bomb was for multiple motivations. The one that is usually cited first, "security," is mostly straightforward, but probably insufficient by itself. In its basics, it is that China, a nuclear state, was to their east and they had fought a war with China was recently as 1962, two years before China detonated their first bomb. "Prestige" is another motivation, and as a prominent member of the Non-Aligned Movement, eager to establish its technological modernism in the wake of its colonial experience, this played a strong role with Indian politicians who since 1945 flirted with the idea of going nuclear (the Indian Atomic Energy Commission, created in 1948, was explicitly designed to keep a bomb open as a possible option). Lastly, there are internal politics to India itself that are looked to for explanations — a government may undertake such a program as a form of political maneuvering.

India's development is unusual in that their first detonation, in 1974, was meant to demonstrate that they had nuclear capability, but they explicitly disavowed weaponizing the technology, saying it only existed as a "peaceful" nuclear device (e.g., for digging holes). Nobody found this especially convincing but it was a way to avoid being overly threatening in this respect. This effort understandably spurred Pakistan, whose conventional forces were greatly out-numbered and out-spent by the Indians, and who had been in various wars and skirmishes with the Indians, to look into their own nuclear option. Factors like prestige and internal politics played a role with Pakistan as well (the Pakistani people are today still quite proud of their nuclear capabilities — in what other area can Pakistan really claim to be one of the forefronts of the world?). As far as we have been able to tell, Pakistan was probably in possession of nukes by the late 1980s. Both India and Pakistan "came out" as military nuclear powers in their tests of 1998.

In all stories about why a country gets nukes, there are local differences and nuances — I have tried to hint at a few in the case of India here, but they are not unusual in that a lot of things had to come together for the nuclear program to become "real." Nuclear weapons are non-trivially expensive and resource-intensive endeavors, they carry high potential diplomatic costs, and they carry high political and military risks, so while it is not at all uncommon for a country to look into a nuclear option on paper (lots of countries have done this), actually committing to building one is relatively rare (only a handful of countries have done this).

On India's program two useful books are George Perkovich's India's Nuclear Bomb (University of California Press, 2002), and Itty Abraham's The Making of the Indian Atomic Bomb (University of Chicago Press, 1998). On Pakistan, there is basically only one really solid work out there at the moment: Feroz Khan, Eating Grass: The Making of the Pakistani Bomb (Stanford University Press, 2012).