It’s a very basic question, and there are many approaches to take. The Ramban on Vayikra 1:9 discusses various approaches we will analyse, and is widely considered one of the fundamental parts of the Ramban’s commentary on the Chumash.
The Ramban quotes the Rambam in Moreh Nevuchim as the first approach. The Rambam writes there that the idea of Jews using animal sacrifice was necessary for the people as they needed a physical method of worship, having been a part of the pagan culture in Egypt and other such places. To battle and rectify the idol worship in the world, the Jews would do the same action for a sacred purpose.
The Ramban disagrees strongly with this on many facets. If we base an entire method of service to Hashem on the actions of fools and sinners, why would Hashem gain anything from it at all? Vayikra 1:9 says that the korban creates אשה ריח ניחוח ליהוה – Hashem “enjoys” the fact that we bring korbanos. The implication of the Rambam is that the korbanos are more for man than Hashem, but if the korbanos were for man, why would Hashem enjoy it? We must find a suitable explanation for bringing korbanos that also explains why Hashem instructs it of us, rather than why why we ought to do it.
The Ramban points out that if the Rambam in Moreh Nevuchim is correct, that Jewish animal sacrifice only exists to battle idol worship, then we would not find instances of korbanos before an instance of idol worship. But this is not so – Adam was the first human – there were obviously no other people around to worship idols – yet he brought korbanos nonetheless, and so too with Noach; his family were the sole survivors of the Flood – so again, there could be no idol worshippers – and we find that nonetheless he did bring korbanos. How would the Rambam explain these instances where there was no idol worship to fight?
He further asks why the solution to idol worship would be to do the same thing in a different way – this seems to lend credibility to the idolatry the korbanos are trying to fight, chas v’shalom. It would seem that it would be better to just eat animals and not have sacrifices at all if we were indeed trying to fight the credibility of idol worship, as eating them shows we don’t consider them to be worthy of special attention.
R’ Yakov Minkus explains the solution to this issue. The Rambam in his magnum opus, the Mishneh Torah (Hilchos Me’ila 8:8) writes explicitly that the yesod – the distilled, fundamental, reason to bring korbanos is a חוק – there is no reason to do it other than the fact we were told to. The Moreh Nevuchim explains the inverse of this – once the mitzva exists, there is a spillover effect that we can relate to more, but the underlying reasoning remains a חוק. With this knowledge at hand, of course Noach could bring a korban, and the question about the non-existence of idol-worship falls away. Battling idol worship isn’t why there are korbanos as a starting point, rather, it helps explain it after the fact.
With this knowledge of the Rambam’s true approach to korbanos, we can suggest an answer to the question of why the countering of idol worship would take a similar form, rather than denigrating it, by simply eating all animals regularly, without any sacrifices at all.
Korbanos have their blood sprinkled on the Mizbeach, by a kohen, in the Beis Hamikdash. The Korban Pesach had none of these key functions, so why is it called a korban at all? R’ Moshe Shapiro explains that the key to understanding this issue is that idol worship is not nothing. Paganism and idol worship have a כח הטומאה – they usurp and corrupt spirituality. Eating an animal doesn’t battle the the negative of idol worship, it just nullifies it. The nullification does not require the Beis Hamikdash, or sprinkling of blood by the kohen. This is why the Pesach could be brought publicly in Egypt. The Rambam in Moreh Nevuchim was saying that all korbanos have this nullifying the negative aspect to which we can relate, but we now see this isn’t the full spectrum of his opinion.
The Ramban then offers a second approach. There is an intrinsic good on man’s part in having korbanos. The idea of a korban is that a person should see the animal as being his substitute, and really, he ought to be sacrificed. The animal takes his place and atones for his sins, and this is the reason to have korbanos.
There is a mighty flaw with this approach too however. Most korbanos are donated, rather than obligated of people, so the Ramban’s approach doesn’t explain the existence of donated korbanos at all.
The Ramban offers a third solution, that is beyond the scope of this site to explain properly. The word קרבן, the root of which is the word קרב, means “closeness”. Offering a korban engenders closeness with Hashem. This is a difficult concept to explain, let alone understand, but to illustrate: we perform mitzvos to emulate Hashem’s ways, but we are not emulating Hashem by bringing korbanos – we are doing something else: we are interacting with Hashem. We are provoking a reaction in Hashem, as the pasuk says; “אשה ריח ניחוח ליהוה” – on which Rashi remarks “נחת רוח לפני, שאמרתי ונעשה רצוני”. This is difficult to illustrate, but there is a difference between doing Hashem’s will, and making it. When we bring a korban, we bring more of Hashem’s will into the world. One could suggest there is an element of creation here.
The Ramban brings a proof from Isaiah 60:7 that says: יַעֲלוּ עַל רָצוֹן מִזְבְּחִי וּבֵית תִּפְאַרְתִּי אֲפָאֵר – the Mizbeach is the expression of Hashem’s will.
So in bringing a korban, a person intentions are going to correlate to how they have extended G-d’s will in the world. This is why there is a concept of pigul, (a lengthy concept regarding what happens in the event that all the actions of a korban were carried out correctly, but someone in the porcess was thinking about something mundane, like the weather. Around 40 pages of Meseches Zevachim are devoted to this) – because the physical animal isn’t what matters – there is a transfer of spirituality here, from potential/theoretical to physical in this world. It’s a very big deal. The improper thoughts mean one can’t interact with what he’s trying to, and the korban has served it’s purpose, as the whole idea is not the physical at all.
So in answer to why we bring korbanos: there is the simple Moreh Nevuchim approach that we are counteracting paganism, the Ramban’s simple approach that we can atone our sins, and the Ramban’s esoteric Kabbalistic approach. We can suggest though, that perhaps the חוק aspect that the Rambam referred to was this third approach, and perhaps all the opinions harmonise together. Admittedly, this doesn’t answer why we bring korbanos, but it does explain what the function of the korban is.