This threat is delivered by the Gamemakers, who are in control of the Hunger Games. They use it to manipulate Katniss into playing the role of a "star-crossed lover" with Peeta, which they hope will boost viewership and increase the entertainment value of the Games.
Here's what the Gamemakers tell Katniss:
* "You know what happens to those who defy us, Katniss. You know what happens to their families."
* "We can make them suffer, just as we can make you suffer. Play the game, Katniss, and they will be safe. Displease us, and they will pay the price."
This threat is extremely powerful, as Katniss loves her family deeply and would do anything to protect them. This is why she chooses to play the role the Gamemakers have assigned her, despite her own personal feelings and beliefs.
This warning is crucial to the story as it:
* Explains Katniss's actions and motivations during the interview.
* Highlights the cruelty and manipulation of the Capitol.
* Sets the stage for the rebellion that will eventually erupt.
While the threat is never explicitly stated on screen, it's a powerful and unspoken force that shapes the events of the story.