How can humans (and veterinarians) benefit from examining this kind of animal behavior? what "new" avenues of thinking are opened up by the butterfly experiments?

Jaap de Roode is a biologist that featured on TEDTalks so the question links to him.

When they have an upset stomach, both dogs and cats will often eat grass to induce vomiting (self medication).

The risk in using man-made medicine when treating certain pathogens includes a tendency for the pathogens to develop into a resistant strain.

Just like humans, the monarch butterfly sometimes gets sick thanks to a nasty parasite. Monarchs survives with milkweed; their caterpillars eat milkweed plants only (Asclepias spp.), and monarch butterflies need milkweed to lay their eggs.

But biologist Jaap de Roode noticed something interesting about the butterflies he was studying - infected female butterflies would choose to lay their eggs on a specific kind of plant that helped their offspring avoid getting sick. How do they know to choose this plant? Think of it as "the other butterfly effect"- which could teach us to find new medicines for the treatment of human disease.

Explanation:

This leads scientists to think of improving Medicine so as to raise offsprings which are resistant to the parasites and diseases that killed their parents.