Leviticus 1:14-17 meaning

Not every Israelite was wealthy enough to offer God a bull or a ram. God's provision for the poor of His people was to accept turtledoves or young pigeons as equally as He accepts the ram or bull as a burnt offering to atone for sins.

The Talmud (a record of Jewish oral tradition) makes the point that each sacrifice has equal impact before the LORD, as each is said to create the same impact, a soothing aroma:

"It is said of a large ox, 'A fire-offering, a sweet savor'; of a small bird, 'A fire-offering, a sweet savor'; and of a meal-offering, 'A fire-offering, a sweet savor.' This is to teach you that it is the same whether a person offers much or little, so long as he directs his heart to heaven."
(Menachot 110a)

If the offering to the LORD is a burnt offering of birds, the priest was to wring off its head and offer it (the head) up in smoke on the altar. Next the blood (of the body) is to be drained out on the side of the altar. This was probably the northside of the altar, the place of the blood (verse 11). Then the priest shall also take away its crop with its feathers and cast it beside the altar eastward, to the place of the ashes. The rest of the feathers would have remained on the bird. A Jewish midrash (interpretive commentary) on Leviticus 1 says:

"But surely you will not find even a common sort of man who can smell the odor of burnt feathers without being disgusted with it! Why, then, does Scripture say that it shall be offered with the feathers? In order that the altar should appear full up, as it were, and adorned with the sacrifice of the poor" (Leviticus Rabbah 3:5)

Finally, the priest would tear it by its wings, but shall not sever it. While the burnt offerings from the herd and the flock were cut into pieces to possibly facilitate even cooking, the bird was to remain in one piece as it would burn just fine as is and would likely not be eaten. Instead of a sacrificial knife as used with the herd or the flock, the priest would use his fingernail and hands to tear or cleave the bird, but not completely in two. Then he shall offer it up in smoke on the altar on the wood which is on the fire; it is a burnt offering, an offering by fire of a soothing aroma to the LORD.

Mary and Joseph were of the poorer class because they offered birds at Mary's purification in Luke:

"And when the days for their purification according to the law of Moses were completed, they brought Him up to Jerusalem to present Him to the Lord ²³ (as it is written in the Law of the Lord, "EVERY firstborn MALE THAT OPENS THE WOMB SHALL BE CALLED HOLY TO THE LORD"), ²⁴ and to offer a sacrifice according to what was said in the Law of the Lord, "A PAIR OF TURTLEDOVES OR TWO YOUNG PIGEONS."
(Luke 2:22-24)

Joseph and Mary were being obedient to the following instructions in Leviticus, 1,500 years after they were given:

"When the days of her purification are completed, for a son or for a daughter, she shall bring to the priest at the doorway of the tent of meeting a one year old lamb for a burnt offering and a young pigeon or a turtledove for a sin offering."
(Leviticus 12:6)

Two verses later it says:

"But if she cannot afford a lamb, then she shall take two turtledoves or two young pigeons, the one for a burnt offering and the other for a sin offering; and the priest shall make atonement for her, and she will be clean.'" Leviticus 12:8

Joseph and Mary took the route of the poor who could not afford a one year old lamb. Ironically, they had the Lamb of God in His first year there with them, yet His sacrifice and ascension ("oleh") would come 32 years later.