Saving face in general involves getting out of a bad situation, one in which you are certainly going to have to face a bad result, without looking too bad. It means that you may have had a bad result, but you maintain your dignity and most of your good reputation and image in the eyes of other people.
In the context of the Vietnam War, this can be related to the idea of “peace with honor.” By the time President Nixon took office, it was becoming more and more clear that the war would not be won. Therefore, Nixon and Kissinger looked to “save face.” They did not want to simply give up in a way that would make America look bad in the eyes of the world. They wanted some way to make America retain its “face” or its honor. For this reason, they tried to achieve victories that would make the US look as if it were withdrawing by choice, not being forced out in defeat.