The book of Exodus describes how the Israelites were delivered from Egypt, and came into fellowship with God. Leviticus describes how that fellowship with God was to be maintained.
v1-2: The Tabernacle was where God spoke to His people, Ex 25.22. As with many other instructions, God spoke to Moses, so that he would explain them to the people.
The first offering was the burnt offering. This was not introduced here, for both Noah and Abraham had made burnt offerings (Gen 8.20; 22.13). This was to be an offering of worship to God, and was therefore the highest offering of the five described here (chapters 1-5). The offering was of livestock, from the herd of cattle, or from the flock of sheep or goats.
v3: There was real cost in making this offering, as the worshipper had to bring a male bull (v3), or sheep or goat (v10), or turtle dove or pigeon (v14). Losing a male also meant the loss of future offspring. The animal to be offered had to be without blemish (as in Heb 9.13-14; 1 Pet 1.18-19).
The offering was a free will offering of worship. Even so, it had to be offered God's way.
v4: The substitute, to make atonement, to guarantee acceptance; Christ died fulfilling this sacrifice, to provide covering/atonement, that we might be accepted before God. Only in Christ can we be accepted.
v5: The worshipper killed the animal, and the priests offered it on the altar; see also v11, Heb 7.27. The sacrifice was not pleasant, nor was the Lord's death, yet it was absolutely necessary. The picture here is of Christ's total and perfect devotion to God (Eph 5.1-2), reflecting the fact that the whole animal was consumed (v9).
Worship of God had to be through the priest, just as our worship of God is through Jesus Christ. In the same way, salvation is only through Jesus Christ (John 14.6; Acts 4.12).
v6-8: The wood on the fire, and the pieces of the animal, were arranged in order, as in v12 also. See also 6.12; 24.3,4,8, and Ex 26.17; 27.21; 39.37; 40.4,23; although this was a free will offering, each detail had to be performed as directed by God. This did not impose extra "form", nor exclude spontaneity, for the offering itself was spontaneous; but the worshipper was not at liberty to do what he wanted. In Christian worship, the same principles of worship are to be applied.
v9: The offering was an aroma pleasing to the Lord.
v10-13: When the offering was of the flock rather than the herd, the same procedure was to be followed.
v14-17: For the poor of the land, the alternative offering was a dove or a pigeon. In John 2.16, Jesus, in cleansing the temple area, simply spoke to those who sold doves, since they were providing for the poor, and were not as culpable as those who selling cattle and sheep, or exchanging money.
In the offering after childbirth, there was also provision for the poor, see 12.1-8, fulfilled Luke 2.22-24.