I have often seen Jews saying that prayers and repentance are sufficient for atonement, then why was the animal sacrificial system was started to begin with, and who are the Rabbis responsible for the view that prayers replaced the sacrificial system? The view that prayers substitutes and replaces the blood sacrificial implies that it is an alternate, contradictory way of atonement than the sacrificial system. Can you also explain when this view began?
- The sin offering is for unintentional sins
- When a person repents and brings an animal that is his possession he is expressing that he has "Sacrificed" something of value as a reminder that he should not repeat his sins.
- According to King Solomon the sacrifice of a wicked person is an abomination to G-d. so important thing for forgiveness of sins is repentance, living by the Law's of G-d and doing good.
- If a person could not afford an animal he could bring flour and oil.(No blood needed)
All these points are in the Torah.
Below is one of the many verses where G-d says clearly he needs us to repent and does not need blood to be offered to him.
Isaiah 1:11 Of what use are your many sacrifices to Me? says the Lord. I am sated with the burnt-offerings of rams and the fat of fattened cattle; and the blood of bulls and sheep and he goats I do not want. 16 Wash, cleanse yourselves, remove the evil of your deeds from before My eyes, cease to do evil. 17 Learn to do good, seek justice, strengthen the robbed, perform justice for the orphan, plead the case of the widow. 18 Come now, let us debate, says the Lord. If your sins prove to be like crimson, they will become white as snow; if they prove to be as red as crimson dye, they shall become as wool.
just to clarify, prayer did not replace the sacrificial system (קרבנות), we pray every day for the קרבנות to return (in שמו"ע by רצה וכו ולתפילתם שעה והשב את העבודה וכו), but since we unfortunately can't bring קרבנות we pray/believe that our prayers suffice, i.e. see medrash קרח פ' י"ח that states ונשלמה פרים שפתינו (הושע יד) אמרו ישראל רבש"ע בזמן שבהמ"ק קיים היינו מקריבים קרבן ומתכפר ועכשיו אין בידינו אלא תפלה. This includes -but is not at all limited to- the sacrifices for atonement (which was needed in addition to repentance).
It is interesting to note what is said from the חפץ חיים (I believe he wrote it in his introduction to אסיפת זקנים on קודשים) that we as long as we don't have the ביהמ"ק are kind of "lucky" in that if we sin unintentionally etc. we can get away "cheap" by praying/repenting, and we don't have to spend a few hundred dollars on an animal and travel to Jerusalem to sacrifice it.
The intended way of atonement is by sacrifices. However, it is not always possible to bring a sacrifice, since the sacrifices are dependent on the Temple. (This point is deserving of a question in itself to be properly explained and understood.) It was know to God from the beginning that there would be a time when sacrifices would be unavailable to people, so it was arranged from the beginning that one can also achieve atonement by other means, by prayer and fasting.
We find such an atonement already by Abraham (Genesis 20:7):
וְעַתָּה הָשֵׁב אֵשֶׁת־הָאִישׁ כִּי־נָבִיא הוּא וְיִתְפַּלֵּל בַּעַדְךָ וֶחְיֵה וְאִם־אֵינְךָ מֵשִׁיב דַּע כִּי־מוֹת תָּמוּת אַתָּה וְכׇל־אֲשֶׁר־לָךְ׃
Therefore, restore the man’s wife—since he is a prophet, he will intercede for you—to save your life. If you fail to restore her, know that you shall die, you and all that are yours.”
Also in Joel(2:12-13):
וְגַם־עַתָּה נְאֻם־יְהֹוָה שֻׁבוּ עָדַי בְּכׇל־לְבַבְכֶם וּבְצוֹם וּבִבְכִי וּבְמִסְפֵּד׃ וְקִרְעוּ לְבַבְכֶם וְאַל־בִּגְדֵיכֶם וְשׁוּבוּ אֶל־יְהֹוָה אֱלֹהֵיכֶם כִּי־חַנּוּן וְרַחוּם הוּא אֶרֶךְ אַפַּיִם וְרַב־חֶסֶד וְנִחָם עַל־הָרָעָה׃
“Yet even now”—says the LORD— “Turn back to Me with all your hearts, And with fasting, weeping, and lamenting.” Rend your hearts Rather than your garments, And turn back to the LORD your God. For He is gracious and compassionate, Slow to anger, abounding in kindness, And renouncing punishment.
And many other sources.
Rambam felt that G-d neither needs nor wants sacrifices. It is a concession to human nature. He says that this is also the view of the prophets.
G-d declares: “To what purpose is the multitude of your sacrifices unto me? says the Lord” (Isaiah 1:11). Jeremiah 7:22–23 says, “For I (G-d) spoke not unto your fathers nor commanded them in the day that I brought them out of the land of Egypt concerning burnt offerings or sacrifice. But this thing I commanded them, saying, ‘Obey My voice.’”
We see from here that G-d neither needs nor wants sacrifices, and only "allowed" them as a concession to human needs.
The first rabbis to say that prayer has replaced the sacrificial system was around 70 CE when the temple was destroyed. The ancient rabbis felt that sacrifices were no longer necessary. It is primitive and backwards. Rabban Yohanan ben Zakkai correctly understood that Temple-oriented Judaism, along with its animal sacrifices, had ceased to exist when the Temple was destroyed. He thereby established “Rabbinic Judaism,” which was people-oriented, centered around the synagogue instead of the Temple, and prayers instead of sacrifices.
Rabbi Dr. Binyamin Lau in his book called “The Sages,” writes that Rabban Yohanan ben Zakkai “heralded a radical innovation in the religious life of Jews, affirming the equivalent value of religious life without sacrifices.” In other words, it simply means that ben Zakkai is saying that we don’t need sacrifices.
 See Guide of the Perplexed 3:32.