If only the future is changed, then why would I cease to exist?
That would be because I changed the past.
If my grandma died, then I wouldn't be born. So I would never have been able to go back in time to kill her. If I didn't go back in time to kill her, then she would be alive. If she is alive then I would be alive. If I am alive then I would be able to go back in time to kill her. But if I killed her, then I wouldn't exist...and so on.