Paul uses the story of Moses' 2nd ascent of Sinai to prove three things in order:
1) In vss.8-11, that if the ministration of the Old Testament, which was a ministration of death was so glorious that the face of its minister shone with glory, then how more glorious is the ministration of the New Testament and therefore Paul’s own ministry, whom God hath made one of the able ministers of the new testament (v.6)?!
2) In vss. 12-16, that the inability of the Jews to look upon the face of Moses (v.7) as he gave the old testament law, and the ensuing vail over his face (v.13), bespeak the inability of the Jews to understand the writings of the law of Moses (v.14).
3) Moses is a figure of Jesus Christ. Christ vailed his glory [partially and briefly exposed in the mount of transfiguration in the presence of Moses!] as he came down from God from Mount Zion in the sides of the north, which is why the Jews could not believe on him anymore than they believed Moses (Jn.5:47!). The vail is taken away in Christ (v.14, 16) who rend the vail upon his death (Mt.27:51/Mk.15:38).