A literal veil had to be pulled over Moses' face when he came down off Mount Sinai, after having been in the glorious presence of God, because he reflected something of God's glory. The account in Exodus 34:29 tells us,
"When Moses came down from Mount Sinai with the two tablets of the Testimony in his hands, he was not aware that his face was radiant because he had spoken with the Lord. When Aaron and all the Israelites saw Moses, his face was radiant, and they were afraid to come near him... When Moses finished speaking to them, he put a veil over his face. But whenever he entered the Lord's presence to speak with him, he removed the veil until he came out".
His face was shining so brilliantly, it took a long time for that glory to fade away. Moses, who had asked to see God's glory (33:18), was unaware when he came off the mountain that he was reflecting the divine glory. It helps to know all the details of that event with Moses to then appreciate how Paul applied the lessons in 2 Corinthians 3:7-18. Indeed, without grasping what happened with Moses, we will never understand the application.
A key connecting point is the Tablets of the Testimony (with Moses) and the New Testimony in Christ's blood. Bear in mind the meaning of the old-fashioned phrase, "Last Will and Testament" - final instructions that a person wills to have carried out after they have died. A Testament is a Covenant. Moses brought in the old Covenant; Jesus brought in the new Covenant. But the Testator has to die first before his will and testament can be carried out. Jesus had to die first before the old testament had been completed, with the new testament brought in. Unless a person sees that, they are 'veiled' from seeing the full glory of the new covenant, in Christ.
Further, consider the event of Jesus being glorified - transfigured - up the mountain. Peter and John witnessed that amazing event when Jesus' appearance was whiter than white, with Moses and Elijah standing on each side, speaking with Him. How significant that Moses be there, given the impending switch-over from the covenant with the nation of Israel mediated by Moses (Exodus 34:10) to the covenant with all who would have Jesus as the Mediator of the new covenant (Hebrews 12:24).
Paul was explaining how seeing no further than the law covenant - the tablets of stone - was acting as a veil, preventing God's people from seeing the glory of Christ as the Mediator of this new covenant. When the old law covenant is read, the glory of God ought to be glimpsed, not a list of rules and regulations! Only when people turned in faith to Christ would that symbolic 'veil' be taken away. The law was pointing them to Christ, but if they saw nothing but legal requirements, they would never see the One who alone could lift the condemnation of the law from them.
That is why Paul speaks of a veil covering the hearts of those in the old covenant. It's their spiritual heart-condition, but not a literal veil, a piece of cloth. This is spiritual blindness due to a hardness of heart against the One whom that law was meant to lead them to - for glory! Paul contrasts the glory of the legal covenant with the greater glory of the new covenant, which brings righteousness. The law, on the other hand, can only bring condemnation because it exposes everyone trying to keep it as a law-breaker who will be judged by it. Now the glory of the risen Christ is seen by those who have faith in him, and they, in turn, reflect something of his glory.
