1. To put something / someone out of its / their misery
This expression now has a largely metaphorical application, and tends to turn up in trivial contexts; moreover, it often has slightly jocular overtones.
It is still possible to use it seriously in connection with the killing of an animal; but it might seem inappropriate in a human context.
In your example, the fact that the person in pain uses it himself makes it less inappropriate; but it would probably seem odd to say:
2. Jason was in terrible pain; so we decided to put him out of his misery.
Student or Learner