Our Gemara on Amud Aleph mentions the well-known principal that the consumption of the sacrifices by the cohanim of somehow effectuates the atonement of the owners of the sacrifice.
The Shalah (Aseres Hadibros maseches Ta’anis Torah Ohr 5:14) explains this idea in a novel manner:
Fasting is a primary form of penitence, whose aims are to repair indulgences that were not appropriate by holding back on pleasure. However, there may have been within the forbidden indulgences, sparks of goodness that still need to be expressed through an appropriate positive action. This repair must be accomplished via the Cohanim eating from the sacrifice.
We can also understand this idea on a psychological basis, which I believe often expresses similar truths as mysticism, just in its own languageit. It is difficult to make changes in character and in deed without replacing it with positivity, Meaning to say, in order to make change sustainable, if a person needs to let go of and abstain from certain behaviors or attitudes, it will be more likely to succeed if it can be replaced with something positive of the same psychic weight. For example, if someone starts a diet, he or she should find an activity (exercise, if possible!) that replaces the soothing quality of food consumption. Or say someone wants to kick the smartphone habit, and stay off YouTube or Netflix. If the person does not come up with a reasonable more positive substitute, the boredom and anxiety will make going back the old habit almost irresistible.
Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation 
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