This mitzvah starts in the previous one, the obligation to testify in court (Leviticus 5:1). Let’s deal with a hypothetical “Moishe.” If Moishe is summoned to testify in a court case and he swears that he has no pertinent information when he actually does, verse 6 says he is to bring a sheep or a goat as a sin offering. However, verses 7-10 tell us that if Moishe cannot afford to bring such an offering, he should bring two birds (pigeons or doves) as his sacrifice. Furthermore, verses 11-13 continue that if Moishe cannot afford the birds, he should bring a flour offering. Any of these are adequate to get the job done, provided one is actually bringing the appropriate sacrifice for his “bracket.”
This sacrifice is called a “korban oleh v’yoreid.” Oleh means “ascending” and yoreid means “descending,” reflecting the variable nature of this offering, whose price goes up or down depending on who is called upon to bring it.
Withholding evidence is not the only transgression that calls for a variable sin offering. Verses 2-10 tells us the other cases in which such a sacrifice is appropriate: if one unintentionally defiled the Temple with ritual impurity or ate sanctified meat, and if one swore to do something but failed to fulfill it.
Like all sacrifices, the reason is to make the person consider his misdeeds and encourage him to do better in the future. In these cases, however, God made special arrangements for people to be able to do so in a manner comfortable for their income level. Why did God do so in the case of these particular transgressions? These sins do not involve acts. They involve speaking hastily or improperly, or acting naturally without forethought to one’s state of ritual impurity. We all do dumb things now and then, but we say stupid things and act without thinking all the time. It would be quite the financial burden for some if they had to pay full price for what might potentially be a more common occurrence.
This obligation to bring a korban oleh v’yoreid as needed applies to both men and women at a time when the Temple service is in effect. It is discussed throughout the Talmudic tractate of Shevuos, as well as in Kerisos. It is codified in the Mishneh Torah in the tenth chapter of Hilchos Shegagos and it is #72 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos.