My wife summarizes it well: "Sometimes, saying 'sorry' is not enough"; i.e. you have to mean it. In the long list of increasing punishments that the Jews could receive if they should reject G-d's statutes (Lev. 26:14-41) even confessing one's sins, and those of one's father, will not be enough, causing G-d to work on behalf of the enemies of the Jews. But, Lev. 26:42-45 tells us that once we are "humbled" then G-d will remember His covenants with the Patriarchs. See also Deut. 28:15-69; 30:1-10. However, G-d reminds us (at Lev. 26:43-45) that no matter what, He has no intention of breaking His covenant with the Jewish people and they can restore their status with Him at any time.
Moreover, Talmudic commentaries on the laws of bringing Temple offerings, whether they be the grain offerings (mincha) or blood offerings, gave special importance that the person bringing the offering, and the kohen (priest) accepting the offering have the proper intent. If not the offering was deemed pigul -- disqualified, and therefore forbidden to be eaten. Scripture's plain meaning (see Lev. 7:15-18), would seem to count as piggul only those offerings not brought at the correct time. The rabbis, however, something within the commandment that went beyond the plain meaning in scripture. As Maimonedes summarizes the rabbis (in his Hilkhot Pesulei Ha-mukdashin 13:2-3):
It was learned from oral tradition that the verse in Scripture, "And if any of the flesh of the sacrifice of his peace offering be eaten at all on the third day" (Lev. 7:18), refers only to where there was an intention at the time of the offering that some of it will be eaten on the third day; and that the same law applied to any sacrifice... if there was an intention that they be performed after their proper time the offering was deemed to be piggul.
In the case of an offering, however, where the intention had not been improper and its blood had been sprinkled upon the altar as required by law, but part of it remained after the proper time for eating it – that part which remained was called "leftover" (notar), and it was forbidden to eat it, but the offering itself had already been accepted and effected atonement.
So the Jewish sages were stricter than the Torah here in that they added a requirement of intent, but more lenient in that leftovers, although forbidden to be eaten after the designated time, were not piggul if the intention at the time of the sacrifice was just and appropriate. It might seem that the Rabbis deviated from scripture, but in full context with other verses of the Torah and the prophets, their ruling is actually consistent. G-d did not want offerings for their own sake, but He wanted offerings that reflected the humility and acceptance of blame by the people bringing the offering. Offerings were not meant to be an excuse for a barbecue or an opportunity for one to show off his wealth. Rather, as Isaiah points out G-d wants His people to fear Him and obey Him -- including His laws of sacrifices -- if one does so with a sense of righteousness. Isaiah 50:10 - 51:1.