פּירוש אויף ויקרא 2:1

ונפש כי תקריב AND WHEN A PERSON (or “A SOUL”) WILL OFFER — Nowhere is the word נפש employed in connection with free-will offerings except in connection with the meal-offering. For who is it that usually brings a meal-offering? The poor man! The Holy One, blessed be He, says, as it were, I will regard it for him as though he brought his very soul (נפש) as an offering (Menachot 104b).

And when a soul offers a meal offering. Who typically volunteers a meal offering? A poor person. The Holy One, Blessed be He, said: I consider it as if he offered his soul (Menachot 104b). For this reason, it states in Parshat Tzav (6:9) it shall be eaten as matzah, meaning poor man’s bread, as all this relates to the poor person who is naturally humble like matzah dough that does not rise.
In this way, the order of all these sacrifices can be explained. Whatever is first in the verses is first to lead to sin, as it is written Jeshurun grew fat and kicked (Deuteronomy 32:15). Anyone who is greater than his fellow in wealth or honor has a stronger evil inclination, as evidenced by what is said When a leader sins (Leviticus 4:22), where the word when [asher] indicates certainty, because his position of leadership brings him to sin as his evil inclination is greater than others.
Therefore, Scripture first mentions cattle, which is presumably offered by the wealthy, then sheep, which is typically brought by the middle class, then bird offerings, which the poor person typically brings, and finally the meal offering, which is brought by those in extreme poverty.

AND HE SHALL POUR OIL UPON IT AND PUT FRANKINCENSE THEREON. 2. AND HE SHALL BRING IT TO AARON’S SONS. “This teaches that the pouring of the oil and the mingling of it together with the flour [of the meal-offering] is valid if done by a non-priest. TO AARON’S SONS THE PRIESTS; AND HE SHALL TAKE OUT HIS HANDFUL. From the taking of the handful [for the altar] and onwards is the duty of the priests.” This is the language of Rashi. Now we have to explain that the duty of the priests does not begin with the taking of the handful, for bringing the meal-offering near [to the altar] precedes the taking of the handful, and that too is invalid when done by a non-priest, as He said, and he shall present it unto the priest, and he shall bring it nigh unto the altar,¹⁹³Further, Verse 8. and afterwards it says, and the priest shall remove from the meal-offering the memorial-part thereof,¹⁹⁴Ibid., Verse 9. which is the handful. Thus you see that it is the priest who brings the meal-offering near the altar, to the south-west corner thereof,¹⁹⁵See Ramban further, 6:7. and after that he removes the handful [to be burnt on the altar]. So also we have learned [in the Mishnah]:¹⁹⁶Kiddushin 36a. “The acts of laying hands upon the offering, the wavings,¹⁹⁷These laws applied to certain animal offerings as well as certain meal-offerings. See Menachoth 61a. bringing [the meal-offering] near [to the altar], and removing the handful are performed by men and not by women.” In explanation thereof the Rabbis said [there in the Gemara]:¹⁹⁶Kiddushin 36a. “bringing near [can be performed only by men] because it is written, And this is the law of the meal-offering: the sons of Aaron shall bring it¹⁹⁸Further, 6:7. — the sons of Aaron but not the daughters of Aaron.” This being the case, the bringing near [of the meal-offering to the altar] is the duty of the sons of Aaron. But the intention of the Sages in saying: “From the taking of the handful and onwards is the duty of the priests” [as quoted by Rashi above], is to say that from this taking of the handful mentioned in this verse and onwards, is the duty of the priests, but not these things which Scripture mentioned here as preceding the taking of the handful, since He mentioned already in this section pouring the oil [and mingling it with the flour], putting the frankincense thereon and bringing it to the priest, and [the priest’s] removing the handful. However, bringing it near to the altar is not mentioned here, [but is stated further on in Seder Tzav 6:7] that all those things stated here before the taking of the handful — namely, pouring the oil and mingling it with the flour, and bringing it to the priest — are valid if done by a non-priest.