After every encounter in the Tent of Meeting, Moses came out with his face alight. Targum Pseudo-Jonathan on Exodus 34:35 says plainly: the sons of Israel saw the countenance of Mosheh, that the glory of the form of Mosheh's face was shining. And then Moses replaced the veil.
The Targum's verb sequence matters: saw, then replaced. The people did not beg Moses to cover himself. He did it on his own initiative, again and again, every time he stepped outside the tent. The rabbis understood this as an act of pedagogical mercy. Moses knew what too much light does to eyes that have not been prepared for it. He had lived for forty days inside that light; the Israelites had not. The veil was his gift to them.
There is a second layer in the Targum's wording — the glory of the form of Mosheh's face was shining. Not Moses was shining, but the form of his face was shining. The light was not Moses himself. It was something that had settled upon him, a loaner from the Shekhinah. The rabbis later used this to argue that holiness is never an intrinsic human property; it is borrowed, and the one who carries it most purely is the one who never mistakes it for their own.
Tradition says Moses was the most humble man on the face of the earth (Numbers 12:3). The veil is proof. A less humble prophet would have walked through camp radiant every day, collecting awe. Moses covered up.
The takeaway: the people who carry the most light are usually the first to dim it for others. Humility and holiness in Judaism are the same gesture — the hand that draws the veil back over the shining face.