In this chapter, the people celebrate their first Passover outside of Egypt, and the cloud of the LORD's presence guides the people through the wilderness.
The first thing I noticed when reading this chapter is that the timeframe is one month earlier than the census from chapter 1. The Passover is celebrated on the fourteenth of the first month, while the census was taken on the first of the second month. This chapter doesn't tell us when the LORD spoke to Moses, except that it was during the first month and we can presume it was before the Passover began (on the fourteenth day).
What this shows us is that the author is not following a chronological story, which is consistent with much of what we have seen before. There are many small stories and vignettes in the Pentateuch that are undated, so in theory they could have occurred at nearly any time during the wilderness journey. What is pretty clear to me is that the author is more interested in grouping and organizing by topic than by chronology. When discussing Genesis 1, I said that the author is "clearly most interested in presenting the creation and coronation of man," and that Genesis "is written in a pre-modernist mindset."
Adherence to a strict chronology also seems like one of the principles of the modernist mindset, which emphasizes a progressive overlay of facts, eventually drawing conclusions from those facts. The author of the Pentateuch seems more interested in a progressive overlay of topics, which he supports by inserting the relevant stories. For instance, there is a brief reference to the Day of Atonement in Ex 30:10, while the Day of Atonement is only really defined in Lev 16. This isn't an inconsistency per se, but it is definitely confusing if you are reading the Pentateuch expecting everything to be properly defined before it is referenced.
The author is adhering to a general chronology (from the patriarchs to the exodus to Mount Sinai...), but as with here, laying out a chronology doesn't seem like the overarching goal. I don't want to push this point too aggressively, because it's a complicated subject that is only loosely related to this chapter, so I'll stop here.
Moving on, the core issue of this chapter is the reconciliation of the Passover with the ceremonial purity laws of Leviticus. There is no provision that specifically states that one must be ceremonially pure to celebrate the Passover, it is implied by verses such as e.g. Lev 7:21. Also, Num 5:2 would suggest that unclean people must dwell outside the camp, while we presume that the Passover is celebrated within the camp.
Anyway, I'm not sure why Passover is the only festival mentioned here. Passover is the first Jewish feast and arguably the most important. Also, verse 13 establishes that if any person is capable of celebrating the Passover but doesn't, that person is to be "cut off", meaning either killed or exiled. This is consistent with the original purpose of the Passover, sparing the firstborn of Israel from death and protecting them from the plagues of God's judgment. If the Hebrews cease to celebrate the Passover, it would be like departing from that divine protection.
One wonders why the only cause of defilement that is called out in this chapter is being near a dead body (being on a journey is not a source of impurity). I've heard various theories, but it seems like the consensus is that if there is a source of defilement which can be avoided, then you are expected to avoid it and it's your own fault if you don't. By this theory, the "dead bodies" would refer to people who suddenly die, such as with the Nazirite vow in Num 6, or alternatively it refers to Israelites who have to perform important burial rites. But there are other sources of unavoidable defilement, such as skin diseases, giving childbirth or continuous bleeding. I presume these would get rolled into the same provision, with the caveat that many people with such conditions would not be ceremonially pure even into the second month. I can't imagine these people would be punished for what are mostly unalterable conditions, so they would probably not celebrate the Passover at all but not be punished for it.
The second half of this chapter discusses the guiding presence of the LORD's cloud by day and fire by night. We had already seen this cloud guiding the people starting back in Ex 13:21 and again in Ex 40:34-38. In particular, Ex 40:36-37 establishes very nearly the same content as this chapter, which is that the people would move out when the cloud arose, and by implication, the people would camp when the cloud would settle.
We see in both Exodus and here that the cloud is closely associated with the tabernacle, which ties it with the LORD's presence. Here it is simply called "the cloud", but in Ex 40 it is called the "cloud of the LORD" and is closely associated with the glory filling the tabernacle whenever the cloud descended upon it. So the cloud is a direct manifestation of the LORD's presence.
In Exodus, the main function of the cloud is to separate the Egyptians from the Israelites (and by extension, protect the Israelites from harm) and to further dampen the effects of the Sinaitic weather. Namely, it is a cloud during the day to block out the harsh sunlight, and a fire at night to help keep the Israelites warm in the frigid cold desert air. Deserts are a challenging environment to survive in, and the Israelites have spent several generations in the relatively stable Nile valley and delta, so they would have been largely unprepared for the journey ahead. Traveling in such large numbers, they would almost instantly consume the scarce desert resources of water and food and be left totally at the LORD's mercy for their provision.
Here the cloud of the LORD is most strongly associated with the LORD's guidance and its movement is considered the "command of the LORD". Not only does the lifting and settling of the cloud tell the people when they should travel, it also settles down in specific locations to tell the Israelites where they should camp. This chapter further emphasizes the literal obedience of the people as they follow the "command of the LORD through Moses". Since the LORD is now issuing commands to the people, to an extent I think this heightens the militaristic tone of Numbers. This, and the people's military obedience to the formation of the camp and marching order, is a great start but as we will see, the overall tenor of Numbers is generally the disobedience of the people.
Like other parts of the Pentateuch*, this section is remarkably verbose. It essentially repeats the same thing over and over for about 10 verses, repeatedly emphasizing the commanding presence of the LORD and the diligent obedience of the people to their King.
One last minor point is that verse 22 says that "whether it was two days or a month or a year that the cloud lingered over the tabernacle..." Unless this statement is hypothetical, it seems to imply that the Israelites would sometimes camp in the same place for a year. Otherwise there would be no basis for the author saying that they obeyed when the cloud stayed for that long. It could be a hypothetical (i.e. "if the cloud stayed for the same place in one year, then the people would have stayed too"), but from the phrasing it does not seem like it is hypothetical. If it is not, then this seems like an allusion for the 40 years of journeying in the wilderness that is to come. The Israelites possibly camped in the same place for a whole year during that journey, though we are never explicitly told so.
*Consider, for instance, the vast repetition between Ex 25-30 and Ex 35-39, which even individually were very generous in detail.