What happens depends on the specific edge. Let's take Trickster as an example:
Trickster: When an enemy model spends Karma to re-roll within 3", this model can spend 1 Karma to steal initiative after its activation. This model must be able to act in order to use this ability.
In the case of this edge, the trigger is "When an enemy model spends Karma to re-roll within 3" ". The activation is "this model can spend 1 Karma" and the effect is "to steal initiative after its activation". So, if there was a trigger and the player activated it, after the enemy activates, this model can be activated (its owner can make an Activation roll to try to activate it). The requirement is "This model must be able to act in order to use this ability", that is, if the model has already been activated or is inactive for other reasons, it cannot activate this edge.
Basically, when you steal initiative, you interrupt your opponent's sequence of activations. Each edge indicates when this can be done and, if a model is identified as a catalyst for that edge, it is it that must be activated if it is capable of being activated.
Is it clearer now?