Why did Moses wear a veil in the Bible?

Answer 1

Moses wore a veil after he came down from talking with God on Mount Sinai because his face was glowing from having been so close to God's glory. The people were disturbed by his glowing face and asked him to wear a veil until the glow faded. You can read about it in Exodus 34:29-35.

Answer 2

Because when he came down from the mountain after talking to God, his face was literally shining or glowing, and it scared the people. Exodus 34:29-35 And it came to pass, when Moses came down from mount Sinai with the two tables of testimony in Moses' hand, when he came down from the mount, that Moses wist not that the skin of his face shone while he talked with him.

And when Aaron and all the children of Israel saw Moses, behold, the skin of his face shone; and they were afraid to come nigh him.

And Moses called unto them; and Aaron and all the rulers of the congregation returned unto him: and Moses talked with them.

And afterward all the children of Israel came nigh: and he gave them in commandment all that the LORD had spoken with him in mount Sinai.

And till Moses had done speaking with them, he put a vail on his face.

But when Moses went in before the LORD to speak with him, he took the vail off, until he came out. And he came out, and spake unto the children of Israel that which he was commanded.

And the children of Israel saw the face of Moses, that the skin of Moses' face shone: and Moses put the vail upon his face again, until he went in to speak with him.