When you stay the course, you continue to do what you have been doing all along. No matter how difficult or time consuming the task may be, you keep at it till you complete it. You stand firm in pursuing your goal. So what the American President is saying that the U.S. must stay in Iraq till there is some semblance of peace in the country.
*Listen Rajesh! Giving up smoking is not going to be easy. You must stay the course. *The diet my wife has put me on is killing me. But I need to lose weight. So I have decided to stay the course.
According to some scholars, the `course’ refers to the racetrack. In this case, it’s the horse that stands firm to reach the finish line — hopefully in a winning position!