This is one of those questions that seems easy but is deceptively hard to answer for a few reasons. One is that there are very few sources on the early South African program that are totally reliable —  figuring out when they decided to make a bomb and why is tricky from a source base alone. Separately, we have different theories about why states get weapons; the one you are looking for is called the "security explanation", e.g., states get nukes because they fear threats. But when you peel away the layers you often find other reasons, too — such as a way for the organizations that would be associated with the weapons to obtain extra internal political power.

But in short. The scholarships on the South African nuclear program suggests that it was largely a side-effect of nationalist Afrikaaner domestic policies that encouraged developing of an independent method of uranium enrichment and keeping the door open for a bomb. The actual decision to make a bomb happened in 1977 as a result of a UN resolution banning the sale of arms to South Africa — part of their becoming a truly isolated "pariah" state. By the late 1970s the South Africans had been alienated from the United States for their apartheid policies, and there were efforts by the Soviets and Cuba to encourage Communist revolutions on the African continent (i.e. Angola and Mozambique). The grand total of this was a decision that it might be wise to have a small number of nuclear weapons on hand, though it doesn't appear clear that there was some kind of direct need or intention for their use.

As Purkitt and Burgess sum it up:

Timing also played a role in South Africa's decisions to develop secret nuclear weapons. South African scientists demonstrated the feasibility of uranium enrichment at the same time that the country began to experience increased threats from enemies in the region and throughout the world. Thus, a confluence of events over time figured into the decision-making calculus of key senior political, military, and nuclear power officials. As the country developed the capability to produce sophisticated nuclear weapons and an increasingly sophisticated defense industry, South Africa's political leaders faced growing threats to political stability from domestic opponents and independent nation-states in the region. Faced with growing threats and the capability to build weapons of mass destruction, South Africa's political leaders opted to build six nuclear weapons.

There are many factors that combined in 1977 — this is just skimming the surface of a complex political period.

As for the public reaction — the program was incredibly secret, but there was a controversy when it looked like they might be trying to prepare a test site in 1977. I don't know what the public reaction was to that in particular (the South African government denied it vehemently), but the program itself was a very carefully kept secret until after its dismantlement.

Two useful sources: Peter Lieberman, "The Rise and Fall of the South African Bomb," International Security 26, no. 2 (Fall 2001), 45-86, and Helen E. Purkitt and Stephen F. Burgess, South Africa's Weapons of Mass Destruction (Indiana University Press, 2005), chapter 3.