The question is a good one, but some of the underlying suppositions need to be challenged in order to answer it.
The Sacrifice of an Animal Cannot Atone For One Man
In Leviticus 6, the animal's life is not enough to atone for the man's life. In some of the in-between verses which the question does not directly reference (4 and 5), the man must make retribution for what he has done to the humans he has sinned against:
If he has sinned and has realized his guilt and will restore what he took by robbery or what he got by oppression or the deposit that was committed to him or the lost thing that he found or anything about which he has sworn falsely, he shall restore it in full and shall add a fifth to it, and give it to him to whom it belongs on the day he realizes his guilt. (ESV)
Not only does he need to bring an animal sacrifice; he must also more than restore the damages he caused. This indicates that the animal sacrifice itself is not enough.
But further, if the man were to sin thus again, the life of the former animal already taken does not atone for this his new crime; he must sacrifice again. This indicates that the life of the animal is at best a partial atonement for only one sin.
The Sacrifice of an Animal Cannot Atone for One Sin
However, if the sacrifice of the animal atoned even partly for one sin, then why do we find in the prophets that God is so angry with the people that he tells them to stop sacrificing entirely—says that their sacrifices and incense and festivals are abominable to him?
"What makes you think I want all your sacrifices?" says the Lord.
"I am sick of your burnt offerings of rams
and the fat of fattened cattle.
I get no pleasure from the blood
of bulls and lambs and goats." (Isaiah 1:11, NLT)
It is not a good sign for the atoning power of the life of an animal if that sacrifice itself can become abominable to him by the weight of sin. Note that this is not a complaint that the people have continued sinning but not brought sacrifice—it is a complaint that they have continued sinning and have brought sacrifice! And as it is, he sees them as entirely unclean.
The Sacrifice of Many Animals Cannot Atone For Many People
Indeed, read any of the prophets; the wrath of God was hot against Israel, even when she brought him many sacrifices. Only two animals were sacrificed on Yom Kippur?—but how many of those days occurred during the lifetime of a man? Those animals also did not atone for the nation, or they would not have to be offered year after year. Moreover, rather than seeing this as a small number of animals atoning for a large number of people, remember that Yom Kippur did not cancel out the sacrifices required by Leviticus 6—another words, on this particular day, a small number of animals were sacrificed; but that did not save the lives of the thousands upon thousands of other animals who still had to be sacrificed day after day.
So how can the sacrifice of one animal atone for many? It cannot.
The Sacrifice of Animals Can Symbolize Atonement
But wait—the law of Moshe still talks about atonement. So what's going on here? David says that God does not desire sacrifice, but a broken spirit—and yet, he does command sacrifice for atonement in the law.
Animals atoned by type—by teaching a sacrificial principle. They atoned by way of shadow; signposts to some other atonement. They taught both by their native symbolism, and by their inadequacy. This is the only way we can harmonize the law and the prophets on this topic.
What is Symbolized?
Yes, "life for life" is the principle of God's immutable justice:
Show no pity: life for life, eye for eye, tooth for tooth, hand for hand, foot for foot. (Deuteronomy 19:21, NIV)
But that is a principle of match crime and punishment. But if the crime merits the death of a man, who does the death of a beast atone? That is hardly a match. However, by this principle, no mere man's life could atone for the whole people either, for that would be a mismatch of justice; for no man's life is worth more than another's—thus the matching principle: there is no respect of the lofty above the lowly in such a principle!
Atonement could only be found in someone whose life is supremely valuable, so that he could ransom the people of God—certainly not in animals, whether a small number or a great number.