In this chapter, the people are commanded to bring their sacrifices to the tabernacle and reminded (again) to not eat blood of any animal.
This chapter contains two separate laws. The first is a prohibition against sacrificing animals anywhere but before the tabernacle, and the second is a re-statement of the prohibition against eating blood (first mentioned in Gen 9:4).
First, we are told that all oxen, lambs and goats must be brought to the doorway of the tent of meeting before it is slaughtered. This is a rather general prohibition and seems to include both "religious" and "secular" slaughter, i.e. killing the animal for its meat. There are a couple of key points here.
The first is that while oxen, lambs and goats are not the entirety of clean animals, they are the central backbone of the Israelite pastoral economy. So, by this rule it would be okay to slaughter like... locusts or something (trying to think of a clean animal that isn't a bull, lamb or goat) wherever you please. However, as should be obvious, the Israelites did not expend a lot of effort herding locusts. They really did spend most of their time and energy shepherding cows and goats and lambs, or else farming. So in practice, this prohibition is nigh binding on all animal sacrifice.
The second point is that there is a presumption the Israelites were sacrificing these animals anyway. Verse 5, which explains the purpose of this command, states that otherwise the Israelites would be sacrificing these animals in the open fields, and verse 7 makes it clear that while in the open fields, they were sacrificing to other gods, the "goat demons" as the NASB puts it (Hebrew "sa'iyr", probably not related but very similar to the Greek "satyr").
This is one of the big reasons why we should minimize the distinction between "religious" and "secular" slaughter (my third point), because in practical terms, there isn't much of a "secular slaughter" concept in the text here or later in the bible. In the vast majority of cases, all animals are killed in a sacrifice, and every feast (wherein many animals are killed) is inherently religious. We have already seen three annual feasts commanded in the covenant, the feast of Passover, Harvest (also known as the first fruits, Weeks, or Pentecost) and Ingathering (Ex 23). These festivals combine their religious function with a large feast, typically involving many animal sacrifices.
Because this chapter is primarily concerned with animals sacrificed for food, it only mentions the "sacrifices of peace offerings" (v. 5), which is the one offering where the meat is given to the offerer to consume.
Fourth, probably the most important issue here is preventing the Israelites from offering sacrifices to "sa'iyr", the goat demons. This is probably part of the broader "God is trying to build a new nation" theme that I emphasized in both Exodus and Leviticus. The natural complement of the Israelites' ignorance of the LORD is their contemporaneous devotion to idolatry (as in Ex 32) and other gods. This chapter, then, is a sort of extension to the first commandment, to have no other gods besides the LORD.
Fifth, the priest is in control of the sacrifice to ensure that all of the biblical regulations are followed: namely, pouring out the blood and burning the fat. These are important biblical regulations, with the fat (as the best part of an animal) always given to God, and the blood (as the life of the animal) always poured out, either on the ground or at the base of the bronze altar. The sacrifices must be overseen by a priest who is professionally trained in the administration of sacrifices that follow these rules (i.e. Kosher).
Sixth, the punishment for not following this law is harsh, generally being "cut off from among the people", which means at least exile, possibly death. This is appropriate when you consider that the people were otherwise offering sacrifices to goat demons, so the main thrust of this law is to prohibit feasts dedicated to idols or any god besides the LORD.
Seventh, a major (side-) effect of this law is the centralization of Israelite society around the tabernacle. While there are many laws and rules that mandate the Israelites to occasionally return to the tabernacle, this is probably the most constrictive because meat from these three types of animals would have constituted a substantial portion of the Israelite diet. When I have been reading through the Pentateuch, I have been asking myself, "would it have been reasonable to follow this law if I were living 50 miles away from the tabernacle?" In many cases the answer is yes, but I think here it is getting much harder. From this, we can read two things. First, the Israelites were probably not geographically dispersed when this law was written. It simply would have been impractical to follow. Second, this law makes it challenging to live further away, inhibiting any future dispersal. Of course, one could simply disobey the law, but only under the threat of severe punishment. As a corollary, this law also substantially empowers the priests, who are now the sole authorized gateway to slaughtering and eating the primary species of herd animals.
Since the tabernacle is the center of worship for the LORD, I think the effect of forcing the Israelites to live (relatively) closer to the tabernacle is an intended side-effect. This in turn would have a side-effect of keeping the people metaphorically "closer to the LORD" and more likely to stay true to their nation's founding faith. In practice, we will later see that the tribes which settle closer to the tabernacle tend to be the ones that stay truest to the covenant. This will be discussed more in Joshua through Kings.
One last point I would like to make before moving on is that according to Lev 3, the latest you can eat meat from a peace offering is the second day after it is slaughtered. By the third day what is left must be burned with fire. Since every offering at the tabernacle must be made as a peace offering, the Israelites are implicitly prohibited from ever storing meat long-term, i.e. dried; unless I'm misreading this somehow, I believe that's what is implied.
The second law of this chapter is a renewed prohibition against eating animal blood. Eating blood has been outlawed many times in the OT already, starting all the way back in Genesis 9 when Noah is first authorized to eat meat at all. At that time, the main motivation for not eating blood is the sacredness of the life that blood embodies. "You shall not eat flesh with its life, that is, its blood." In this instance we are told something different, that the prohibition is connected with atonement: "I have given [blood] to you on the altar to make atonement for your souls; for it is the blood... that makes atonement."
This statement is also connected with the lifeforce, as in v. 11: "the life of all flesh is in the blood". In a certain sense, we could say that it's not the offering of blood that atones for sin, it is the offering of life that atones for sin upon the altar and before the ark of the covenant. The sacrilege here is to eat the life of another animal, because as Genesis explains, "Surely I will require your lifeblood; from every beast I will require it. And from every man, from every man’s brother I will require the life of man." To consume flesh is allowed, but to consume life is not, because all life comes from God, and to God every life is to return. God takes an accounting of life and demands an account, a return of all life back to himself.
Lastly, we are given a provision that one may eat an animal that was found torn by beasts (without bringing it to the tabernacle for sacrifice), but because one would have no opportunity to completely drain the blood, one became ritually impure until evening. I'm a bit surprised by this allowance, but I wouldn't say it's contradictory. Also, this seems contradictory to Ex 22:31, so I don't know why there is a ceremonial provision now.
And with that, we will move on to everyone's favorite topic, laws prohibiting immoral sexual relations!