If an elderly person breaks a hip then develops a blood clot in the lungs, how would a blood thinner save his life?

2nd clot: If that person lives long enough to start a blood thinner (ie, they don't immediately die with the first blood clot), the issue becomes preventing a second, fatal clot. That's where treatment with blood thinners are important: preventing the second, fatal clot. Moreover, the first clot will gradually dissolve on blood thinners over time.