how can you conclude that the animal was heaving heavy groans?

The First Lord is talking about a deer that was wounded by a hunters arrow. The First Lord says he saw the stag rest beside Jaques. He says it was groaning in pain, "The wretched animal heaved forth such groans That their discharge did stretch his leathern coat."

"that their discharge stretch his leathern coat" what does this mean...?

Breathing so hard their skin was stretched.