A major creepy thing for some was a complete lack of background music.
If you're young enough to have first watched it ten years or better after the fact, long after the Cold War was over, you really aren't going to grasp the full impact of the film. On first run television during the time period, the US was neck-deep in very real nuclear fear. Not only was the culture awash with apocalyptic and post-apocalyptic imagery, the news was on constant alert to what the Soviets were doing at any moment and every credible expert was saying that the odds were very likely that nuclear war was at some point inevitable. two entire generations grew up believing that they would not survive to adulthood, and that unlike the post-apocalyptic fiction common at the time there would be no survivors. The film wasn't presented as what "might" happen, it portrayed what everyone believed would happen and did so in brutally factual fashion. And not only did it grip the nation with its vivid depiction of what everyone was already imagining and terrified of, it actually convinced U.S. President Ronald Reagan to overrule the hawkish military establishment and begin diplomatic overtures to the Soviets. That's right, it was so terrifying it actually caused a change in foreign policy so profound that it might have directly prevented the horror it depicted. That's not just nightmare fuel, it's thermonuclear grade nightmare fuel.
Some were freaked out by the sight of their home city being vaporized.
The scenes in which the nuclear bombs make people turn into skeletons. The effects may be a little hokey, but considering the nature of the film it's incredibly effective.
Perhaps even more nightmarish than The Day After was Testament, made around the same time. It took place in a single town and focused on the effects a nuclear war had on a small group of people. The intimacy is powerful, especially when it came to the main character nursing her children through radiation sickness and watching them die one by one.
The scene of the attack itself. Nuclear explosions ripping the sky, electricity going out the huge blasts... it's no coincidence that when the film premiered, there were no ads after the attack. It was planned by ABC to be shown that way. Commercials were shown up through the climatic events that climax with the nuclear exchange. No commercials were planned after the war, to avoid breaking the Willing Suspension of Disbelief of a wrecked world the producers were trying so hard to create.
Shortly before the film aired, a misunderstanding in Germany led to extremely heightened tensions, which some consider to have come even closer to nuclear war than the Cuban Missile Crisis. Yes, the original audience was extremely lucky that they were watching the film, rather than living it.
Around that same time, the Soviets almost pulled the trigger as well, due to a malfunction in their radar network that falsely showed five ICBMs inbound to Moscow. It was only thanks to a Soviet colonel on duty who realized that the United States, first-strike policy or no (and we did have such a policy, to help perpetuate MAD by making the Soviets think twice about playing chicken), would not be so suicidal as to fire a mere five warheads as their first strike and correspondingly did not pass the alert up the chain of command that World War III may have been averted.
The last message of the film states its events was far from what would be likely during an actual nuclear exchange. It was extremely optimistic.
The images of the immediate aftermath of nuclear war. It can be permanently scarring to see the farmers, at the very end, looking out over their ash-covered fields, covered with dead livestock and fallen corn, that hasn't even started to rot even weeks after death — because radiation has even killed the flies and bacteria that would start decomposition — wondering how the hell they're even going to grow a crop to feed the survivors.
Survivor Joe Huxley repeatedly tries to contact other survivors with a shortwave radio, but gets no responses. The final lines of the movie are him broadcasting again: "This is Lawrence. This is Lawrence, Kansas. Is there anybody there? Anybody at all?" It is truly chilling.
A woman is unable to feel any joy over the impending birth of her child, knowing that she will likely die before her child is fully grown, her child will likely have a drastically-reduced lifespan, and the years they do survive will be hellish. When the doctor advises her to stay hopeful, she responds
Hope for what? What do you think is going to happen out there? You think we're going to sweep up the dead and fill in a couple of holes and build some supermarkets? You think all those people left alive out there are going to say, "Oh, I'm sorry. It wasn't my fault. Let's kiss and make up"? We knew the score. We knew all about bombs, we knew all about fallout. We knew this could happen for forty years. But nobody was interested.
A character is horrified when he learns the military is executing looters, thieves, etc. without trial, but later passes by a firing squad execution and doesn't even blink. The nightmare fuel isn't the execution, but rather what it illustrates: the breakdown of society. The legal system has become largely irrelevant and by necessity people may need to resort to behavior that would have been unacceptable before the bomb. They might have to rethink the most basic ideas about the value of human life (for instance, euthanasia may become widespread/accepted because of severely limited medical supplies and food). Essentially, the survivors have to learn to live in a whole new world, with different rules.