The story
By analogy with an actual jaw and the fact that this mechanical device allows the possibility of preserving the life of the victim being rescued.
Etymology from Wiktionary, CC BY-SA 4.0.
jaws of life
Today it means: Emergency rescue equipment used to open a severely damaged passenger vehicle with a strong mechanical jaw function that forcibly pushes or pulls metal components apart, to quickly and somewhat safely extricate the trapped occupants. Full definition →