The significance of this story lies in what has just happened to the Duchess. She has just been arrested by Bosola in disguise and is going to be taken to her palace by guards. The story she tells is therefore a kind of parable that refers to her own state as a captured individual. The story of the dog-fish and the salmon and how they interact and the moral that can be drawn from it applies to her own situation. Consider the following excerpt from this story:
Our value never can be truly known,
Till in the fisher's basket we be shown:
I' th' market then my price may be the higher,
Even when I am nearest to the cook and fire.'
So, to great men the moral may be stretched;
Men oft are valu'd high, when th' are most wretched.
The Duchess, having just been arrested, is "valued" more highly because of her "wretched" state, but she is appealing, like the salmon is appealing, to be identified as just another human who is worthy of being shown mercy and kindness. The story thus represents a plea for a common humanity in spite of everything else that has happened in the play.