In this chapter, we return to the priesthood with laws governing ceremonial cleanliness and who can make offerings.
The first thing we are told in this chapter is that priests (excluding the high priest) are governed by a set of restrictions about when they can become ceremonially unclean. Remember that as before ceremonial impurity is not a sin, but because the priests "present the offerings by fire to the LORD" they are held to a higher standard than the general populace. As we have previously read, being near a dead body results in ceremonial impurity, which even includes burying or mourning a dead relative. The priests are restrained from doing this unless the person is in their immediate family (note that a married sister is no longer immediate family, because she becomes a part of her husband's family).
This is a surprising and rather severe prohibition, because mourning for deaths in the family was a pretty big deal in the ancient Near East. For instance, we saw that Joseph and the Egyptians grieved for seventy days over the death of Jacob (Gen 50:3). Ancient Egypt is generally renowned for embalming and entombing their kings within the world famous pyramids, as well as the great emphasis within their religion upon death and the afterlife. The afterlife is not broadly discussed in the OT, but there are several deaths in the OT which are widely mourned, with the death of Jacob being probably the best example. Also see Gen 23:2. We will see many more later in the OT.
There are two types of restrictions here. The first is that the priest cannot go into the presence of the dead body, because that would result in ceremonial impurity. The second restriction is against the religious or cultural practices of the local Canaanite nations, which are expressed in verse 5: shaving hair or cutting oneself. These customs had already been condemned for all of Israel in Lev 19:27-28, but it's condemned again as part of the broader subject of mourning the dead. As before, it is prohibited as part of the principle of separation, dividing Israel from the surrounding nations.
There is no earlier prohibition of marrying a divorced woman or widow, so this is an added rule that is unrelated to any earlier commands. In addition, this chapter outlaws prostitution by a priest's daughter. I believe prostitution is generally outlawed under Lev 19:29, and it is arguably outlawed under Deuteronomy 22:13-21 and Ex 22:16 as well, by implication. So this seems like it's mostly a restatement of earlier law, with the added command that a priest must not marry a prostitute.
These rules are only a few out of the many ways to become ceremonially impure, with some others being skin diseases, having a release of bodily fluid (blood, semen, etc.), or being put in contact with another person who is ceremonially impure, and none of those ways are forbidden here. (Of course, many of them are involuntary and unpreventable.)
The high priest is restricted even further, so that he cannot mourn in general (v. 10; uncovering the head and tearing your garments is an expression of grief) and cannot approach the dead bodies of anyone, including close relatives. While being the high priest is a role with great power, it is a difficult position to walk so close to the holiness of God.
As I have pointed out before, the standards of behavior in the OT becomes more and more strict as one operates closer to the LORD. There are things that it is OK for the people of Israel to do, but not the priests, and the high priest is constrained the most of all. There are things that are OK for the foreigners in the land to do, but not the people of Israel (for instance, eating unclean animals). At the same time, there are laws that also bind on foreigners, because simply by living on earth everyone is near to the LORD in certain respects.
At the same time, the high priest has more access to the tabernacle than anyone else, because he alone is authorized to enter into the most holy place behind the veil. The priests have access to enter the tabernacle outside the veil, while laypeople do not. The people of Israel can partake of the Passover and other ritual festivals, while foreigners cannot (unless those foreigners become circumcised and join the covenant). So there are restrictions, but each added layer of restrictions comes with yet greater access to the LORD and to the holy things associated with the covenant. While there is provision for foreigners to join the covenant, there is no provision for voluntary ascension to the priesthood: it is a permanent, hereditary and mandatory life-long assignment for the men in the family of Aaron. To be a priest and enter the holy place, one must be chosen.
The second half of this chapter prohibits any son of Aaron from ministering before the bronze altar or within the tabernacle if that priest has any notable physical deformity. This prohibition is conceptually similar to the requirement that most sacrifices be "without defect" (for instance, see Lev 1:3, 1:10, et al.). The idea then, and the idea now, is that the people of Israel are expected to offer the LORD their very best. To sacrifice a defective or sick animal would be treating the sacrificial system like a garbage disposal, which is precisely the opposite of the point. No one may appear before the LORD empty-handed (Ex 34:20), because the offerings are supposed to represent the vast blessings the LORD has given the people. The offerings must be without defect for the same reason, to personally acknowledge to the LORD that he has not given the people "defective blessings".
This passage might seem discriminatory, and perhaps in some ways it is. But there have already been many discriminations made: between Israelite and foreigner, between the sons of Aaron and the people, between men and women. The priesthood is not an ordinary occupation; these people are required to stand before the LORD and intercede on behalf of the people through prayer and sacrifice. If you look at it from the perspective of "defective priests", i.e. sons of Aaron with blindness, disfigurement, etc., then yes, this looks like unfair discrimination. But the priesthood is not a job, the priests are representatives of the nation before their divine sovereign, and so the people are responsible for how they present themselves to their lord.
As noted in the text, priests with defects are nevertheless allowed to eat the holy foods (i.e. the priestly share of the sacrifices offered at the tabernacle), which provides a source of nourishment and support, but they are not allowed to represent the people before their king. It is notable that only the ceremonially pure may eat the holy food, so this shows that disfigurement is not a source of impurity or uncleanliness. Still, my understanding is that many parts of the sacrifices are given to the priest who offers the sacrifice, so not being allowed to make offerings would in that sense impair a man's ability to acquire food.
Lastly, this is the first chapter of the bible that uses the phrase, "food of his God" to describe the sacrificial offerings (v. 6, 17). Lev 22 has the last reference, so it's only in these two chapters that the expression appears. This is an interesting expression because it connotes a sense that the various offerings (which are all food-related) are sort of like "feeding God", as if he eats the offerings. I think that's evidently not the case. I hypothesize that all of the offerings are food-related because the vast majority of Israelite effort went towards growing food, whether crops or livestock, and making offerings related to their work is a fairly natural extension. It is for this reason, and not because God has some peculiar need, that they make food offerings. I'm not going to do a proper etymological study, so consider this an exercise for the reader, but I will note that the word translated "food" (NASB) is Hebrew "lechem", which means all of bread, food and meat. So it's a very broad term for anything related to food.