To me, the concept of saving face means that the country must maintain its reputation. It cannot be seen to be weak. It cannot be seen to have suffered defeat because this would ruin its prestige in the world.
As the Vietnam War dragged on, it became more apparent that it would be very hard to win the war. However, the US doggedly kept trying to be more and more aggressive. It kept sending more troops to the country and resisted the idea that the war was not worth fighting. Much of this came about because of the need to save face.
At many points, the US had declared its commitment to maintaining the “freedom” of South Vietnam. Presidents like John F. Kennedy had done things like promising to “bear any burden” to prevent the spread of communism. In this way, it could seem that national prestige was on the line in Vietnam. If the US withdrew without winning, it would look like its word was no good. It would look like it could be defeated. This would make it look weak in the eyes of the world.
This, to me, is the meaning of saving face. The country must not look like it has gone back on its word or has lost a war. It must try, to maintain its prestige even at a terrible cost.