In my fantasy world, a massive kaiju-like creature awakens wreaking havoc about the empire. So, in a desperate attempt to preserve life, the royals and a few lucky bunch take refuge in a massive underground cave system. This lasts for decades, and the first batch of underground kids are whipped up. The government does not want to tell these children of the massive creature roaming around above their heads as to create a new generation in which the previous mistakes ceased to exist. But like all kids, they are curious, so they would ask their parents and teachers why live in a cave, so how would they answer? I'm not looking for an exact quote, but rather things they could reference to tell them or that children would understand, believe and accept. Note that I would like them to somehow refer to something above ground that makes it too dangerous to leave as it helps with the plot of my story.
I'm not convinced I understand your question but I do have something to say about it so I'm going to post my answer anyway. How would you refer to a creature you don't want anyone to know about? Short answer is you wouldn't. At all.
That said, I think the whole premise is wrong. Living in a cave tells me the population is living under a siege mentality. That is ultimately self-defeating and scientifically couldn't happen. Without sunlight, you can't grow crops et al. Even if you were living on mushrooms or some variant, the only source of nutrition for them would be human excrement which means single pathogens would infect the entire population really quickly. On top of that, it's a closed energy loop without perfect efficiency meaning that eventually the entire population would die of starvation. The only way to solve this problem is to deal with the Kaiju, and that means teaching the kiddies about it so they (as the next generation) can add to the knowledge needed to solve this.
Let's assume though that life is sustainable in a cave at the society level (let's assume some form of geothermal power). Let's further assume that the society doesn't want the kiddies to leave, like ever. The best way to solve this is religion.
You develop a creation myth along the lines of 'Once we were a liberal and licentious people, who roamed far from the sustaining warmth of Mother Earth. We thought we were all powerful and Mother Earth had to teach us the error of our ways by raising the beasts of the Earth against us and teaching us once more our place, which is close to her. Mother Earth sustains us and we are her servants. Do not stray from this place of honour in her presence or she'll remove her sustaining warmth...' I think you get the picture from there.
Ultimately, the way to fight reason and curiosity is emotion and religion has been a particularly powerful tool in that fight in key times in the past and (arguably) the present. This is not meant to be a disparaging comment against religion or even faith; I'm just making the observation that religion is essentially a structured framework for a given society or subset thereof. It imposes a common set of principles and beliefs, and self manages dissent through the 'do this you're good, do that and you're bad' behavioural regulation model. So, use it to your advantage in this case.
It sounds like, at least in this world, the children are at least aware of a world outside of the cave in which they live in. If the motivation is to prevent them from sneaking up to the surface, or just to keep them complacent in continuing life in the cave system, then there must be some reason for them to not go to the surface, and it seems like simply telling them that there is a giant human-killing monster on the surface is out of the question for one reason or another.
Since the reason needs to be very compelling, I would just twist the truth. Tell them if they go to the surface they will die. Maybe the surface has been irradiated, is scorching hot or bitterly cold, or the air is poisonous. Even better if going to the surface risks exposing the rest of the populace to whatever threat is above. Then contrast that image with the ideal cave life, so the eventual reasoning is "why risk checking out the surface when we have everything we need right here?"
You could have an Adam and Eve type myth they tell their children. Perhaps even modelled on the original story but jazzed up a bit to make them the children fear going outside. It might go something like this: When God got angry over the apple scrumping incident in the garden of Eden, he banished the people into the caves as a punishment. He warned the people that they would have to make a new life underground to pay for their sins. Anyone who dared to venture outside ever again would immediately be slain by the gods shooting off thunder bolts and lightening at them.
To ensure that the message sinks in groups of children are taken into some of the upper caves when there is a thunderstorm brewing. They are told that some bad person has gone outside and that bad things are sure to follow. Sure enough shortly after they see the flashes and hear the rumbling from above. They then have to sing, chant or carry out some other act of worship to appease the gods for their sins continually for an hour or so until the gods are satisfied and the thunder and lightning goes away.