What does "to save one's life" mean?

Even if one's life depended on it, as in I couldn't eat another bite to save my life, or Betty wouldn't climb a mountain to save her life. This hyperbolic expression nearly always follows a negative statement that one wouldn't or couldn't do something. Anthony Trollope used a slightly different wording in The Kellys and the O'Kellys (1848): “I shan't remain long, if it was to save my life and theirs; I can't get up small talk for the rector and his curate.”