In this chapter, the LORD commands Moses to perform a census of the Levites, and each family of Levites is assigned its duties related to the tabernacle.
There are four main elements to this chapter. There are two genealogies (or as I like to call them, YAGs: Yet Another Genealogy), there is the assignment of tasks to the Levitical families, the Levitical census, and the association between the Levites and the firstborn sons spared of the Passover from Ex 12.
First, there are two brief genealogies, covering the small family of Aaron and the clans of Levi. I don't have much to say about these that I haven't said before. We have even seen genealogies for Levi before, so this really isn't new material, it's just placed here to thematically tie together the various clans of Levi that are being counted.
Second is the assignment of tasks to the Levitical families, which is intermingled with the census. There are three families of Levites, the Gershonites, Kohathites and Merarites. The Gershonites and Merarites are jointly responsible for the tent of the tabernacle and the courtyard screen (the former family carries all of the fabric and skins, the latter family carries all of the poles and bars and metal bases). The Kohathites are responsible for carrying the holy things that go inside the tabernacle like the table and lampstand and altar, as well as the bronze altar which is outside of the tabernacle. At first glance, this seems like a really small amount of work for such a large number of people, but keep a few things in mind: 1) this includes infants and children who are not associated with the work of the tabernacle, 2) the Levites are given more responsibilities later, 3) the Levites also need to do a lot of work related to their families and raising children, etc, so they are probably not working at the tabernacle full-time.
In terms of power structure, this is a very substantial boon to the tribe of Levi, who already held the priesthood and now handle all of the things of the tabernacle. They are literally encamped on the four sides of the tabernacle, forming a buffer between the dwelling place of the LORD and the rest of the nation. Even though Judah is in many respects the foremost tribe, Levi is also rapidly consolidating a base of power as the tribe most associated with ministry to the LORD, and this is definitely something the other tribes notice. Expect discord to emerge as the other tribes begin to challenge the honored role of the Levites, and as the Levites challenge the honored role of the priests, and as Aaron the priest challenges the honored role of Moses. (Literally all of these happen in Numbers.)
Next I will discuss the census. Consistent with the census in Num 1, the census here also assigns a leader to each family of Levi, with Eleazar (who will become the next high priest) the familial ruler for Kohath. If you recall, Aaron and Moses are sons of Amram, son of Kohath.
The structural form of the census is comparable with the census from before, given that it counts through each family, assigns a familial leader, and then measures the total at the end. Some key differences are that this census is less structured (no joint oversight with the other tribes). Intermingled with the census results are the obligations of each family. But probably the biggest difference is that this census is from one month and up, rather than 20 years, because this is not a military census, this is really more political or religious. Moses is counting the Levites both to apportion their tasks for the sanctuary and to figure out how many firstborn sons can be redeemed. They probably use one month as the minimum age because of the high infant mortality rate of this era. It's a more reliable statistic to count after one month, when you don't have like 10%+ of the children dying anymore. I think that's also one of the big reasons why circumcision occurs on the eighth day, because so many children would die before then.
The 22,000 Levites is not an apples-to-apples comparison to the prior census because this includes men down to the age of one month, while the prior census was only 20 and older. Therefore we can see that the Levites are by far the smallest tribe.
This census is also commingled with the assigned positions for each family in the camp, with the position of honor on the east side taken by Moses and the priests and the three families taking the other three sides.
Fourth, and lastly, is the association between the Levites' service and the consecration of the firstborn after the Passover. We already saw a process for redeeming firstborn sons in Ex 13 by the sacrifice of a lamb, but now in addition to that we see that the LORD wishes to take the Levites into service in exchange for the firstborn of Israel. Of course, Ex 12 and 13 say nothing about the tribe of Levi in particular, so while this may be the justification for their dedication to the LORD, I see the real root cause to be Ex 32:29, when Moses tells the Levites to dedicate themselves to the LORD because of their zeal (and possibly their familial tie to Moses).
Other than that, there's only one other thing I'd like to point out which is the suspiciously low number of firstborn sons. Recall that the total number of males over 20 is just above 600,000, while now we are told the number of firstborn sons over the age of one month is just over 22,000. Even if each family had six sons apiece, you would expect over 100,000 firstborn sons. We also have to deal with the peculiarity that a family must redeem their son with a lamb, and then it is somehow exchanged again for the Levites.
Probably the best explanation I have heard for this yet is based on Lev 27, which establishes the price for redeeming vowed persons. In that case, a male between one month and five years is valued at five shekels, the same price charged here for the 273 excess firstborns. This seems to strongly imply that the only firstborns counted here are less than five years old, possibly because all of the older firstborns had already been redeemed. Of course, the text in this chapter doesn't say that the 22,273 firstborns are less than five years old, but then again, the OT "doesn't say" a lot of things, so this explanation seems pretty valid to me. Also, the author is possibly trying to maintain a sense of parallelism between the "one month and up" of the firstborn sons and the "one month and up" of the Levites (which probably includes all the men of Levi and not just one month to five years). Under my hypothesis, this parallelism is a bit misleading because it doesn't mean the same thing for both groups, but this is the best way I see to get the numbers to line up correctly.
Other explanations generally revolve around implied statistical anomalies amongst the Israelites, such as the death of many firstborn children while enslaved in Egypt or that the Israelites had abnormally many children. Another possible explanation is that the Passover itself redeemed all of the firstborns at the time, and the 22,273 were only the firstborn sons born in the year since then, which would be an abnormally large number of new births. But yeah, I think the first explanation I gave is probably the best.