Leviticus 5:7

But if his hand find not sufficiency to bring a lamb, let him bring, as an offering for the trespass that he hath committed, two large turtle doves or two young pigeons before the Lord; one for a sin offering, and one for a burnt sacrifice.

if he cannot afford [literally: if his arm cannot reach] Since it is with one’s arm that one generally labors, and it is the arm that acquires objects, Scripture uses this metaphor to allude to one who lacks something.

one as a sin-offering and one as a burnt-offering Rabbi Yiṣḥaq said that the reason for the burnt-offering is that, since he could not afford a lamb, perhaps an evil thought entered his head. The more straightforward explanation, in my opinion, is: One provides the equivalent of the fats of a sin-offering that are wholly consumed, while the other is the actual sin-offering that is called for by law.

AND IF HIS MEANS SUFFICE NOT FOR A LAMB. Scripture has been lenient towards these sinners by allowing them to bring an offering of either higher or lower value. It is possible that the reason for the leniency with regard to the offering in the case of oaths [i.e., the “oath of testimony” and the “oath of utterance”, as explained above] is because the punishment is not excision [were they to be done wilfully]. In the case of defilement of the Sanctuary and the holy food [He mitigated the obligation of the offering] because the person who did it erred whilst engaged in performing a religious duty, for the priest who eats the holy food or enters into the Sanctuary to prostrate himself or to bring an offering is engaged in performing a religious duty, and his intention is towards Heaven. Therefore even though he sinned on account of having forgotten his state of uncleanness, Scripture gave him more ways of atonement.
Rabbi Abraham ibn Ezra commented that the reason why one [of the two birds brought by the poor man in the offering of higher or lower value] was a burnt-offering, is so that it should be offered on the altar in lieu of the fats of the animal sin-offering [brought by the affluent person]. He has explained it well.

ואם לא תגיע ידו די שה, “if he is unable to afford the price of a lamb, etc.” seeing that people guilty of entering sacred grounds in a state of ritual impurity or swearing a futile oath, do not derive any benefit from committing that sin, the Torah made allowances for them if they found offering a lamb as sin offering a financial hardship, and offered two alternatives depending on their financial status. No such relief is granted to people who inadvertently ate forbidden fat, blood, or who ate on the day of Atonement, (thinking it was on a different date), or people performing forbidden activities on the Sabbath for their personal benefit. Anyone who had derived personal benefit by making use of sanctified animals or vessels has committed two sins simultaneously, and is also not qualified to avail himself of what is written in this verse. He is viewed as if had tried to steal from G–d. All these people have to pay for the value of the animal offered by the priest on their behalf.