Again his eyes filled with tears and he cried as intensely as before, but this time, not with pain but with joy; he felt a great refreshing sensation spread inside him, extinguishing the hot something that had nearly strangled him.

which best explains how this passage determines the use of metaphor in the story?

You are correct! "His eyes filled with tears...a great refreshing sensation...extinguishing the hot something..." basically means that he is crying out of relief, and the absence of the agony he felt before is "refreshing". The cause of this relief is not solely the outward cause of his torment has been eliminated, but the pure act of crying, allowing his emotions to pour out—as if he is putting out the flame of pain with his tears of joy. The act of crying serves as a final relief to the torment, rather than the actual physical struggle. This is why the tears, not his solution, are the "extinguisher" of his pain. Hope this helps!