Why doesn't North Korea just silently build their nuclear weapons?

You cannot develop nuclear weapons silently because you have to test them, both warheads and missiles, which is very visible from space and by radars.

Other answers have already covered the main point: nuclear weapons work best as a deterrent, which requires visibility. At the point where you're actually using them, everybody has lost.

I just wanted to add that there are solid game-theory reasons why belligerence and irrational-seeming behaviour can be a strong political tactic.

Imagine you're negotiating with somebody who is holding a hostage and has an explosive vest strapped to himself. If you know that he is a rational person who values his self-preservation highly, then blowing himself up is the worst possible outcome - even worse than being arrested and going to prison. That being the case, there is no scenario in which it makes sense for him to detonate the vest, so you can effectively ignore its existence. At which point you can just send in a few police to grab him.

However, if he is an irrational person, there is a risk that he might detonate the vest if provoked. Assuming you know this, then it gives you a good reason not to just try to grab him - which boosts the chances that he'll get some outcome better than going to prison.

Hence, even if he is in fact a rational person who wants to save his own skin, there can be an advantage in appearing irrational and volatile. KJU may well understand that nuclear war with the USA is a losing proposition, but it is in his interests to give the impression that he believes otherwise.

It is easy to see North Korea make statements aimed at the United States or another country (e.g. South Korea or Japan) and assume that the intended audience of the statement is the US (or whomever). But in many cases, the intention is to speak to a domestic listener.

Such a domestic listener might be the general public or a popular general. To that listener, Kim Jong-un has to make statements that are consistent with the domestic image that he is projecting. If internally he is claiming to have secured nuclear weapons that make North Korea the equivalent of China on the world stage, then he has to act like that. Sometimes that means that his bluster outruns his capabilities.

In order to justify the army, Kim needs an external foe. So he has to claim that the US (or whomever) is a danger to North Korea. Even if that's nonsense, that's what he needs to claim to perpetuate his military budget. But if his military budget is to do all these wonderful things to protect North Korea, then people expect protection. Otherwise, why would people labor to preserve his administration.

To that end, sanctions help him. They show that the international community is out to get North Korea. They separate North Korea from other countries. Yes, they make North Korea poorer, but they leave it more united in its poverty. However, if Kim left the people time to think, they'd notice that things weren't getting better. So he has to keep moving. Otherwise the people would want to see progress in their every day lives. And that's difficult to do without real internal changes that would loosen his hold on the government.

There are a few reasons I can think of why the DPRK would not want to keep their nuclear ambitions secret:

1. Military Dictatorships are inherently unstable. Kim Jong-Un has almost as much to fear from his own family and generals as he does from anyone outside of North Korea. If he appears to be strong locally, there is much less chance for a coup. This is one of the reasons why the fairy tales that are popular among North Korean people are encouraged, though I imagine the number of true believers in the country is probably much less than what his regime would like. Having an external threat to point to can have a stabilizing effect by keeping those who may wish to depose him isolated out of fear.

2. In prior times, signs of belligerence from North Korea was sometimes met with some form of international aid. If you give a bully your lunch money once, they're likely to try to take it again.

3. When it comes to nuclear weapons, if you build some and no one knows about it then the concept of mutually assured destruction does you no good. It's not enough to have them, people must know that you have them and that you are willing to use them.

4. The current American administration has been particularly bellicose against North Korea with its rhetoric. The regime may wish to deter a potential first strike by appearing strong, and reminding people that they could start something akin to World War III if they wanted.

Because there is no deterrence if no one knows about it.

So why not wait until you have it? A poor, poorly developed country isn't going to make rapid, steady progress. If you look at the history of NK nuclear and missile testing, it is littered with many examples of embarrassing, dismal failures. Not having a proven, established and impressive track record in developing a state of the art arsenal is not a deterrent, either.

So they bluster and behave is if they have made that progress, in order to put doubt into minds of others. That, and along with their oversized (for a country with their population and economy) conventional military forces so close to South Korea has to substitute for actual objectively dangerous deterrence until they can get the real thing in place.

What seems to be real is the paranoia they have that South Korea and the USA are just itching to invade and conquer. They also know that, even if they could inflict damage, they could not withstand a military fight with the USA. Though we can play "chicken and egg," being named as part of the "Axis of Evil" by the leader of the lone military super-power in the world, shortly before he invaded one of the other members of that club, probably only helped to confirm (along with the less dramatic but steady rhetoric from the US side of the back and forth), in their eyes, the threat from the USA. Hence the need for deterrence.

Think about the big, powerful dog and the tiny, fluffy dog when they meet in the street. Which one is making the most noise and usually acting aggressively? Usually the smaller one. It's a pretty normal behavioral reaction, when fearful, to try and bluff the larger opponent through exaggerated aggressiveness.