I believe the answer is that really the damager deserves to have his limb actually punished, and the money is a כופר, or way of redeeming his limb by replacing it with money. This is why the verse says:
וְלֹֽא־תִקְח֥וּ כֹ֙פֶר֙ לְנֶ֣פֶשׁ רֹצֵ֔חַ אֲשֶׁר־ה֥וּא רָשָׁ֖ע לָמ֑וּת כִּי־מ֖וֹת יוּמָֽת׃
You may not accept a ransom for the life of a murderer who is guilty of a capital crime; he must be put to death.
but for other physical damages, money is accepted in place of the limb of the damager.
The Rambam actually makes this point explicitly in Mishneh Torah:
זֶה שֶׁנֶּאֱמַר בַּתּוֹרָה (ויקרא כד כ) "כַּאֲשֶׁר יִתֵּן מוּם בָּאָדָם כֵּן יִנָּתֶן בּוֹ" אֵינוֹ לַחֲבל בָּזֶה כְּמוֹ שֶׁחָבַל בַּחֲבֵרוֹ אֶלָּא שֶׁהוּא רָאוּי לְחַסְּרוֹ אֵיבָר אוֹ לַחֲבל בּוֹ כַּאֲשֶׁר עָשָׂה וּלְפִיכָךְ מְשַׁלֵּם נִזְקוֹ. וַהֲרֵי הוּא אוֹמֵר (במדבר לה לא) "וְלֹא תִקְחוּ כֹפֶר לְנֶפֶשׁ רֹצֵחַ" לְרוֹצֵחַ בִּלְבַד הוּא שֶׁאֵין בּוֹ כֹּפֶר אֲבָל לְחֶסְרוֹן אֵיבָרִים אוֹ לְחַבָּלוֹת יֵשׁ בּוֹ כֹּפֶר:
When the Torah says: "If a man disfigures a person, as he has done so shall it be done to him" (Leviticus 24:20), it does not mean to inflict injury on this man as he did on the other, but that the offender fittingly deserves to be deprived of a limb or wounded in the same manner as he did, and must therefore indemnify the damage he caused. Furthermore, the Torah says: "You shall accept no ransom for the life of a murderer" (Numbers 35:31), implying that only for a murderer no ransom is accepted, but compensation is taken for the loss of limbs or for injuries sustained.
So the simple pshat of the passuk actually describes what the damager deserves, but the Torah gave him a way to redeem his limb with money.
EDIT: The purpose of payments for damages is to make the victim "whole", which is why the word for payments "tashlumim" comes from the root שלם, meaning complete. The purpose of physical punishments in the Torah seem to be both for atonement and deterrence. This is made clear in the case of murder by the statement of Rabbi Shimon ben Gamliel in Makkos
סנהדרין ההורגת אחד בשבוע נקראת חובלנית רבי אליעזר בן עזריה אומר אחד לשבעים שנה רבי טרפון ורבי עקיבא אומרים אילו היינו בסנהדרין לא נהרג אדם מעולם רשב"ג אומר אף הן מרבין שופכי דמים בישראל.
A Sanhedrin that executes a transgressor once in seven years is characterized as a destructive tribunal. Since the Sanhedrin would subject the testimony to exacting scrutiny, it was extremely rare for a defendant to be executed. Rabbi Elazar ben Azarya says: This categorization applies to a Sanhedrin that executes a transgressor once in seventy years. Rabbi Tarfon and Rabbi Akiva say: If we had been members of the Sanhedrin, we would have conducted trials in a manner whereby no person would have ever been executed. Rabban Shimon ben Gamliel says: In adopting that approach, they too would increase the number of murderers among the Jewish people. The death penalty would lose its deterrent value, as all potential murderers would know that no one is ever executed.
In the case of a damaged limb, BOTH aspects apply. The victim deserves payment for monetary damages, but the damager also should lose a limb as a deterrent to future damage. The Torah allows him to redeem his limb by making the victim whole financially.
As to how we can interpret the Torah's punishments this way here but not in other places, the ultimate answer is that we can only interpret verses in line with the tradition from Moses. However, in this case we see that the Torah recognized the concept of "kofer", or redeeming the punishment with money by damages to others (such as an ox killing a man) and excluded it by murder, which implies that in other cases of injury it would be permitted. We don't find the concept of "kofer" by other types of sin, so it is not as sensible to assume the concept should exist there. But again, ultimately the rules for interpretation are set by tradition.