Threats were literally extra wounds that the Free Peoples' player was forced to distribute amongst their companions each time a character died.
Threats were generally a way to prevent the Free Peoples' player from throwing in 'chump' companions designed to absorb many archery wounds or go up against massive minions with no real consequence (i.e. sending Pippin to be demolished by the Balrog). Now in addition to losing the companion, additional wounds had to be distributed to the remaining companions.
The Shadow player could add threats using various cards, such as the Morgul Destroyer:
When threats were added, the Free Peoples' player would place wound tokens upon their dead pile (next to their discard pile), and once a companion died they would then distribute that many tokens among their companions.
One caveat: There is a maximum number of threats equal to the number of companions in play.
In addition to being a stellar complement to any Shadow archery deck, there were many cards, such as the Destroyer, that were able to use threats to their advantage, including The Witch King: