The Hebrew word used here (bo) is translated as "within it," but it also means "within him." And the Kli Yakar explains that both after and because of the initial psychological effect upon the sinner, the sacrifice performs an actual spiritual cleansing of the person from within.
And it says, "And the fire of the Altar shall burn within it."
If it refers to a regular fire, it means to say that it "shall burn within him"—meaning, within the person who offers the sacrifice because the fire burns in His Wrath from the point of view that his [the sinner's] whole being is panicked and recoiling backwards as if to say, "What is this I have done? For I have made my deeds evil and forfeited my life and therefore, I am deserving of all these repercussions [dinim]."
And in reality, the fire of the Altar acts within him and causes him to do teshuvah [repentance].
And if it is speaking of the fire of Torah Law [Dat Torah], its simple meaning is that it actually burns within him and incinerates every bad thought and all the animalistic qualities that are within him.
As the Kli Yakar says, all one's bad or sinful thoughts and all the animalistic qualities that drive a person to sin...the Altar's fire consumes them along with the physical sacrifice.
How to Activate that Process Today
This is an incredible Kindness on God's Part.
And while we don't have the Beit Hamikdash, the Mizbe'ach (the Altar), and the sacrifices today, our Sages have emphasized how merely reciting the sacrificial reading (called Korbanot) can have the same effect as actually offering the sacrifices.
(The Korbanot reading can be found in every Orthodox Jewish siddur/prayerbook toward the very beginning, somewhat after the morning blessings and before Pesukei D'Zimra. It starts with Ketoret/The Incense Offering and continues into Korbanot/The Sacrifices. You can find links to the Ketoret and Korbanot readings, either separately or together, here, Hebrew only. An English translation of the reading as it appears in the siddur, starting with Korbanot, is here. But if you want to start with Ketoret, then you can start here.)
Yet the Kli Yakar notes that a significant part of the fire's purifying process was due to the accompanying emotion of the person who offered the sacrifice.
So we can glean from this that while reading the Korbanot, it is important to visualize the scenes described within and also to realize that we deserve to undergo this process, too.
However, Hashem in His Great Mercy, decided to use an animal in our stead - and not just as a superficial tit-for-tat, but to actually cleanse us from within, giving us a perfectly New Beginning that will hopefully lead to success on our spiritual mission and enable us to fulfill our innate potential to the maximum.
Rabbi Shlomo Ephraim of Luntschitz (1550-1619) lived in Bohemia (which is today Poland and Czechoslovakia). He served as rabbi and dayan and wrote several books, the most well-known being his commentary on the Chumash known as the Kli Yakar.
This is my own translation and any errors are also mine.