I grew up in the States but am east Asian by ethnicity/heritage and have some knowledge of east Asian culture (though obviously my parents didn't think too highly of it, otherwise they probably would've made a more concerted effort to educate/indoctrinate me about it).
The concept is quite simple, it's primarily about bolstering external perception in order to promote the reputation of a group that one self-identities with - be that the family, the company, or the country. You define an in-group and an out-group, and within the in-group honesty and transparency is permitted (at least with respect to the domain of the in-group, you're not going to be sharing family secrets with your co-workers, for example). However, when it comes to the out-group, every effort is made to give the appearance that activities within the in-group are efficient, successful, "harmonious" (i.e. lack of conflict between members of the in-group) - in other words, bury all dirty secrets and make everything look utopian, even if it isn't. Transparency is discouraged because it is bad PR, and members of the out-group (i.e. the rest of society) are expected to have lower expectations as to the amount of information that is provided through "official" channels. So in order to obtain such information, members of the out-group turn to gossip, espionage, etc.
I wouldn't say that "Western" culture (I hate that term because I reject the existence of that distinction as philosophically valid) doesn't practice "face-saving" to some degree, it just isn't taken to the extremes that it is in east Asia because of societal expectations regarding transparency and accountability. I for one think that this is one area where people in China, Korea, and Japan can learn a lot from "Western" countries. After all, face-saving is simply an aspect of tribalism, institutionalized.