The intricate halachos of kashering utensils are derived from Parshas Matos (Bamidbar 31:22-23). Upon returning from the battle against the Midyanim, Elazar Hakohen instructs the Jewish people that Hashem had commanded Moshe how to make the food utensils taken in battle permissible for use. Specifically, he taught that there are two different methods of kashering and which method is to be employed depending on how the utensil was used in the preparation of non-kosher food. Based on the pesukim in Parshas Matos, the mishna (Avodah Zarah 75b) teaches us that a utensil used with hot water must be kashered using hot water via a process known as hagala, and those utensils which absorbed the taste of non-kosher food directly through fire must be kashered by using fire, i.e. via the halachic process of libun.
"K'bal'oh kach polto - the way something is absorbed is the way it is removed" is not only the guiding principle of kashering utensils, but also addresses how we "kasher" ourselves as well. When a person sins there is a negative impact on his soul which can only be removed by teshuva. The degree of teshuva depends on how much one's soul has been permeated with sin.
When seeing the Jewish people worshipping the egel hazahav, the Torah tells us that Moshe saw the egel and the people dancing around it and then broke the luchos. Apparently it wasn't the sin of Avoda Zarah itself that warranted the smashing of the luchos. Rather, it was the enthusiasm that was exhibited by dancing that required the luchos to be broken. A sin committed half-heartedly will be easier to rectify than one done with enthusiasm. Just as a utensil must be kashered using physical circumstances that are equivalent to the manner in which it absorbed non-kosher food, so is the way a soul must be cleansed. A misdeed performed with excitement and enthusiasm must be corrected by teshuva that is equally motivated.
The Halacha teaches us that certain utensils cannot be kashered through conventional means. Kli cheres - earthenware - are so absorbent that they can never be completely purged of their non-kosher contents. However, Chazal (Pesachim 30b) teach us that even earthenware can be made reusable by placing it in a furnace (there are several halachic and practical considerations concerning implementing this principle). The Rishonim raise the question as to why placing earthenware back in the furnace is effective; if absorbed taste can never be completely removed, why should placing these utensils in a furnace render them permissible for use? Tosfos (Pesachim 30b) quote Rabbeinu Tam explaining that since earthenware is formed initially in a kiln, returning them to such a source of heat is in effect remaking them. As such, they are no longer the same utensils that absorbed non-kosher taste.
Similarly, occasionally the regular methods of cleansing one's soul are not sufficient, since the absorption of sin is so deep that a superficial teshuva is not effective. Rather, one must resolve to change to such an extent that (s)he become a new person, as the Rambam (Hilchos Teshuva 2,4) describes the baal teshuva no longer being the same person as before.
Utensils purchased from a non-Jew must be immersed in a mikvah prior to use. This mitzvah also teaches us how to purify our souls. The Rambam (Hilchos Mikvaos 11:12) observes that immersion in a mikva is symbolic of immersing oneself in the waters of Torah. The only way to purify our souls is to immerse ourselves in the study of Torah. The halachos of kashering and immersing utensils not only enable us to observe kashrus properly, but serve as a guide for our spiritual growth well.