In this chapter, Moses commands the people to offer their sacrifices at the LORD's dwelling place.
The first part of this chapter is similar to Deut 7:5, which also instructs the Israelites to destroy the religious symbols of the Canaanites. I have nothing to add other than pointing out the similarity.
The rest of the chapter is basically guidance for the Israelites regarding their sacrifices. There are two classes of food: holy and common (I am inventing these terms, but I think the text supports this delineation). The holy food consists of burnt offerings, tithes, vows, sacrifices, the firstborn of the herd and flock, and so on (v. 6). These are all the various classes of religious offerings that we have read about in Exodus, Leviticus and Numbers in various places.
Common food is when the people "desire to eat meat", and they are authorized to slaughter animals "within any of your gates", which means that it must be done within a city.
The holy food (primarily animal sacrifice, but also include the tithes of grain, oil and wine) must be brought before the "place the LORD chooses", the first occurrence of this phrase in Deuteronomy.
I discussed the term "the place which the LORD your God will choose" in my introduction to Deuteronomy, so I won't talk about it here. In summary, it is meant to be the future permanent residence of the LORD's presence, just as the tabernacle now holds the LORD's presence in their wandering.
This chapter is similar in spirit to Lev 17. In that chapter, the LORD commanded the people to bring all their sacrifices to the tabernacle without exception. I remarked at the time that this has a scalability problem: when the Israelites settle into the promised land, some of them will live a considerable distance from the tabernacle and would not be able to bring their sacrifices to it. This chapter directly mentions the distance problem in v. 21. Because when Israel is wandering in the desert, they travel in a relatively compact camp, so distance isn't as much of a problem. It is only when "the LORD your God extends your border as he has promised you" (v. 20) that this becomes a problem.
This chapter addresses the scalability problem by creating a second class of animal slaughter which may be done in any city of Israel, eaten by the clean and unclean alike. It is only the religious offerings that must be taken to the tabernacle. In addition, the Israelites are still prohibited from offering sacrifices in "the open fields" or the high places, because those would be outside of the city gates.
Lev 17 also includes numerous prohibitions against eating the blood of sacrifices (Lev 17:10, 12-14), and this chapter repeats that prohibition several times as well (v. 16, 23-25, 27). My guess is that these two topics (prohibition of blood, the location of sacrifices) are tied together because both of them have to do with the laws of offerings. In addition, the blood is considered sacred and is used for many of the sacrifice rituals (see Lev 1, 3-5 for more on this).
In addition, the Levite "who is within your gates" is also listed as a prescribed beneficiary of the sacrifices made before the LORD. "The Levite" is mentioned three times (v. 12, 18-19) and always in the context of sharing in sacrifices made at "the place which the LORD your God will choose", which means that this only includes the holy food and not the common food that I discussed above.
There's two things that are notable about this passage regarding the Levite.
First, it means that the Levites will travel with the offerer to the "place". Since it is only the "Levite within your gates" and the offerer must travel to the new temple, the Levite is also traveling to the temple to share in the offerings. We already know that the priests are responsible for the offerings, so it's unclear what, if anything, the Levite will do. Also, note how v. 12 and 18 list parts of a man's extended household. First it mentions "sons and daughters", and then servants, who are not family, but part of the shared household. It includes Levites last, suggesting that they are to be supported as part of the household, as the next tier down from servants.
Second, it has the implication that the Levites will be living spread out amongst the cities of Israel. This is strange when you consider that the Levites already have 6 dedicated cities in Israel. At this point, I am genuinely wondering what these Levites will be doing since they are supposed to be in service of the priests and the priests will all be with the tabernacle (or the temple, as the case may be). This chapter is essentially creating more provision for the Levites without explaining what they will be doing.
This chapter doesn't directly relate to any of the ten commandments, so opinion seems to differ on how it can be related back to the general stipulations. Some commentators suggest that it relates to the second commandment by destroying idols, while other commentators claim it relates to the first commandment because it emphasizes the exclusive worship of the LORD at his temple over the polygamy of sacrificing wherever one pleases. I don't have any particular opinion, because as I've said before, the first and second commandments are heavily related. The first commandment (to have no other god before the LORD) is implicit in everything about the covenant and there is nothing that will be entirely unrelated to it.
Moses concludes by warning the Israelites to not adopt the practices of the nations they are going to destroy. Good to know.