Moses was not aware that the skin of his face had become radiant.
G-d Himself chiseled the first set of tablets out of the rocks on Mount Sinai, whereas the second tablets were chiseled by Moses. Nevertheless, it was specifically after receiving the second set of tablets, rather than the first set, that Moses' face shone.
This is because when something is given to us from G-d without our having worked to earn it, it does not penetrate our very being. It is thus no accident that the first tablets were broken, whereas the second set never were. When we work for something, it can remain with us permanently; something that is received unearned can be more easily lost.
Because Moses chiseled the second tablets himself, their holiness could penetrate his physical body, and therefore his face shone. Similarly, the effort we expend in studying the Torah and fulfilling G-d's commandments refines even our physical bodies. If we exert ourselves to the point that the Torah penetrates us, our faces glow.