We don't make threats, but USA gets involved in their affairs, which they do not like.
If you discuss about this situation with East Asians - you will find out most conservative Asian countries dislike foreign involvement in their country's affairs. This applies very closely to China, North Korea, North Vietnam (formerly), and some other smaller countries like Tibet and Cambodia. Right now USA has strong presence in SK and it affects NK because they feel foreign outsider is messing with their personal issue and dislike it.
It is similar with Middle Eastern conflicts. Some of the conservative members of Middle-East dislike US involvement to achieve democracy in Afghanistan, Iraq, Israel-Palestine conflict and so on.
Because of protecting cultural traditions, groups like Hezbollah and Al Qaeda formed by the interest of some people in the country to preserve their viewpoint (whether you think their group is right or wrong, it is their opinion and preference about their native country).
You can say it's same principle - So, that's why NK express the extreme disapproval of USA involvement.