In this chapter, the LORD describes the compensation of the priests and Levites, which they receive instead of an inheritance in the promised land.
NB: To anyone who is following my blog, I apologize for being gone for the past couple weeks. I was traveling internationally and now that I'm back, I will try to make up for the lost posts.
At first glance, this chapter seems like a non-sequitur to the last, because the prior chapter ends with the people fearful for their lives, and this one is talking about the role of the Levites and so forth. If you think about it though, this chapter actually does (indirectly) answer the fears of the people. Remember that Korah's rebellion was a challenge to the Aaronic priesthood, where Korah and the Levites sought to usurp Aaron's role in the religious system. This results in the judgments and the fear at the end of chapter 17.
The LORD's response here is to reaffirm the Aaronic priesthood, first emphasizing the responsibility and then emphasizing the commensurate payments. Importantly, we are taught that the priests are responsible for the "guilt in connection with the sanctuary" (v. 1), because this is what aroused the LORD's anger previously. If the people respect the Aaronic priesthood and follow the proper protocol, they have no need to fear the "guilt" that resulted in the death of Korah and his followers. In conclusion, "there will no longer be wrath on the sons of Israel". (v. 5)
Next, we are told that basically everything which is given to the LORD is at least partially handed over to the priests, as compensation for their service. This compensation is mainly from three sources: 1) most sacrificial offerings, 2) the firstfruits offerings, 3) the redemption of the firstborn. Leviticus discusses the sacrifices and the share given to the priest in great detail. We were told that the Levites were given in lieu of the firstborn, but now that principle is extended to include the sacrifices and redemption for firstborn animals. For the first fruit offerings, I don't believe we were ever told what those went to, but it seems like a natural conclusion that they should be given to the priests.
Later in this chapter we are told that the priests have a fourth source of income, the tithes from the Levites. This is important because the most consistent source of income for the Levites, the sacrifices, must be treated as holy, which means they cannot be eaten in a state of ceremonial impurity (see Lev 22). Since a woman is ceremonially impure for either 33 or 66 days when she gives birth, going without food for that long would be a serious problem. The tithes are therefore a consistent and dependable food source for the priest and his whole family, which is supplemented by the other sources I mentioned above.
Verse 20 concludes that the LORD is "your portion and your inheritance among the sons of Israel". So the priests are given all of these material considerations for their service, and in exchange they do not have any inheritance within the promised land.
Next, we are told that the Levites are given a roughly similar deal. They do not have any inheritance of land, but in exchange they are given a tithe from the people. Similar to the priests, the Levites help bear the service of the tabernacle and protect the Israelites from the LORD's wrath, which is possibly why the Israelites are expected to pay them a tithe. Lev 27 incidentally mentions the cost of redemption for tithes, but does not formally instituted any particular tithe. Here we are told the tithe is the inheritance of the Levites, but again it doesn't seem to be formally instituted. It just says that there is a tithe which is given to the Levites. It's not clear to me if this is a reference to the tithe of Deut 12, the tithe of Deut 14, or some other tithe altogether. All of the various tithes reference the Levites in some way, which leaves some ambiguity about how many tithes were enforced and for what purpose these tithes were used.
What we can clearly tell from here and Deuteronomy is that the nation was expected to materially support the Levites, freeing the Levites to serve the ministry of the tabernacle on behalf of the nation. We are also given some idea of the relative size of the priesthood compared to the Levites, as the Levites are given a tithe from the whole nation (10%) while the priests are only given a tithe from the Levites (1%).