In this chapter, Miriam and Aaron continue the assault upon Moses, challenging his role as the leader of Israel.
As I implied when discussing the last chapter, whenever Moses solves one complaint it only opens the door to a new complaint. In this case, having given the people an excessively large supply of meat, we are now told that Miriam and Aaron are both "speaking against" Moses, ostensibly because of his Cushite (i.e. upper Egyptian) wife, but what they were challenging in v. 2 is whether Moses deserves a position of leadership over the nation given that the LORD has "spoken through us as well".
This is ironic for two reasons. First, Aaron and Miriam both already hold positions of substantial leadership. Aaron is the high priest and has considerable authority. Miriam is much less discussed, but we were told in Ex 15:20 that Miriam is a "n'biyah", derived from "naba", a prophetess. This is probably the basis for why she says that the LORD has spoken through her. Second, in just the last chapter Moses (with the LORD) put the spirit upon seventy of Israel's elders, so it very much seems that Moses is not intentionally holding onto power. Aaron and Miriam choose now, after Moses shares power with others, to seek their own power grab.
I think there are two sides to this issue. The first side views this as an affront. Aaron and Miriam are already two of the most powerful people in the nation, so why are they seeking even more power? You can view this as a case of raw arrogance. The second side is viewing this from the perspective of Aaron and Miriam, who probably see Moses hording all of the power for himself. The first question we have to address is whether that's true, and the second question is whether that is a reasonable perception given what we can guess they know.
The reality, as we are told it by the bible, is that the LORD was supporting Moses 100%. There are many things we don't know about Moses, but if we can trust the biblical text, then we do know that the LORD was the real power behind Moses. Why he chose Moses is something up for debate (which I have debated, back in Ex 4), but we can have no doubt that the LORD called Moses and gave him all of the authority that he now, in fact, holds.
Note that verse 3 is likely a later addition, since it doesn't really fit in the narrative. Of course, there are many things in the Pentateuch that don't exactly fit in the narrative, but this one is more striking than usual. I frequently hear pastors/teachers say with some irony that Moses is calling himself the most humble man on the earth, but with all due respect this verse probably is a later editorial.
To address the second question, in much of what we have read, the LORD is talking to Moses who then recounts these laws or commands to the rest of the people. What does anyone know of the LORD except what Moses says? This does leave something of an information gap between Moses (who speaks to the LORD "mouth to mouth", i.e. directly and intimately) and the rest of the people, who generally only have Moses's words. How much could the people know about the LORD apart from Moses's instructions? What have the people seen that validates Moses's position? I don't want to go back and review everything we have read, but I think the answer is, "a lot". Apart from many other incidents, Moses told the people the Passover ritual which protected them from the very real plague of the firstborn that struck the Egyptians. Every time the LORD performs something in accordance with Moses's words, that is a validation that Moses does speak on behalf of the LORD.
I've already said the Israelites have a memory problem, but I don't think this is the case here. Miriam and Aaron are challenging Moses precisely because they remember the many occasions where Moses has done something great, bringing back laws of the LORD or performing a miracle or entering the tabernacle to speak with the LORD (and where they are forbidden).
Regardless of their reasons or what they thought, this episode shows that the LORD supports Moses fully, as we might have suspected.
For some reason, only Miriam is punished. Whether that's because she was the more energetic challenger or for some other reason is unclear. Notably, verse 1 lists Miriam before Aaron, which is a standard formula to show that she was the leader of the two in this endeavor.
Anyway, Miriam is temporarily cursed with leprosy, which as we saw in Lev 13, makes her unclean and she must depart from the camp. Similarly, in verse 14 when the LORD says "if her father had but spit in her face, would she not bear her shame for seven days?" I'm fairly sure that spit is one type of bodily fluid that results in ceremonial impurity, which creates a two-fold message. The more obvious message is that having your father spit in your face is a mark of rebuke and disgrace, and that is the LORD's intent with giving her leprosy. The less obvious (to us) message is that leprosy, as with saliva, makes her ceremonially unclean so that she must (by law) depart the camp for seven days.
This is a fairly short story, but it shows us that the people are rebelling against Moses even at the very top, while the prior chapter showed us that they were revolting en masse. I think we should be able to see why Moses is having such a hard time with this people.