(6) The Lord said to him further, “Put your hand into your bosom.” He put his hand into his bosom; and when he took it out, his hand was encrusted with snowy scales!
(7) And He said, “Put your hand back into your bosom.” — He put his hand back into his bosom; and when he took it out of his bosom, there it was again like the rest of his body.
(6-7) His hand was encrusted with snowy scales! … there it was again like the rest of his body: Like the staff that became a serpent, the next sign, snowy scales (“leprosy”) on Moses’ hand, is directed both to Moses and to the Jews, to whom God here is instructing Moses to demonstrate these signs.
For Moses himself, leprosy might have been a punishment for speaking leshon hara – which means slander or malicious gossip. Moses has clearly expressed his expectation that the Jews would not believe him, which can be seen as his having slandered the Jewish people (although the translation given at 4:1 is “What if they do not believe me and do not listen to me, but say: The Lord did not appear to you?”, the more literal rendering is the much more emphatic: “But, behold, they will not believe me, nor hearken unto my voice; for they will say: The Lord did not appear to you”).
Be that as it may, the leprosy on Moses’ hand is, at worst, localized and transitory – it quickly passes, because Moses’ words do not actually constitute “malicious slander,” for they are just an innocent mistake. Moses’ fears that he would have difficulty establishing mutual understanding with the people were not completely groundless. But in the end, all the problems were resolved.
For the Jews, however, these signs symbolize Moses’ status as an intermediary standing between God and the people.
Leprosy is a sign of separation, of living in isolation from people (“Being unclean, [the leper] shall dwell apart; his dwelling shall be outside the camp”, Lev. 13:46). But separation can be either negatively or positively motivated. Kings, for example, live separately from the people. Moses, too, throughout his life, from childhood and through old age, is largely separated from the Jewish people. He grew up in Pharaoh’s palace, then lived in Midian for an extended period, and remained distant from the people even after the Exodus. Moses will climb Mount Sinai while the people will have to wait below. Moreover, Moses’ tent will stand apart from all others (33:7).
Thus there is always a divide between Moses and the people. We might even say that Moses is a “half-leper” (half, because He nevertheless remains always in control; that is, a certain amount of contact between Moses and the people is maintained, just as his hand can be either affected by leprosy or fully healed of it).
Moses’ isolation from the people is one of the reasons that the Jews do not immediately believe him. But the sign of the half-leprous hand is meant to show the Jews that this isolation is positively motivated. Moses does truly stand between the Divine and the human, between two worlds. He is the medium through which the Divine teachings are transmitted to the people.
 See, for example, the story of Miriam, Num. 12:10.