This idiom captures the deep concern Japanese have for maintaining face. “Face,” of course, means one’s positive image, one’s public identity and correct behavior within the community. The “having no face part of the expression can be interpreted as “not knowing what expression to wear,” or “not knowing even how to compose one’s expression” when having to face someone at a time when one feels deeply ashamed.
A: Sumimasen. Konkai no purojekuto ga umaku ikanakatta no wa zenbu watashi no fuchuui ni yoru mono desu. Mattaku awaseru kao ga arimasen.
A: すみません。今回のプロジェクトがうまくいかなかったの は全部私の
A: I’m sorry. It was due to my carelessness that the project didn’t go well. I have no idea how to face you….