When someone kills accidentaly he is sent to live in an ir miklat ("city of refuge") until the Kohen Gadol ("High Priest") dies. This seems like an arbitrary time. What does the Kohen Gadol's death have to do with manslaughter? What's the connection?
A person who killed accidentally had to stay in the City of Refuge until the Cohen Gadol died. This could be 1 day or 80 years.
If a person killed on purpose or in a completely faultless manner (Onnes), he does not have to stay in the City of Refuge until the Cohen Gadol dies. (Rambam Hilchot Rotzeach 6:3)
Killing someone on purpose is black and white, either he did or he didn't. There are witnesses and warning, and there is no uncertainty. Same thing when killing someone in a completely faultless manner, either it was or it wasn't. There is no gray area.
Killing someone accidentally (Shogeg) on the other hand, enters a zone of uncertainty. How much of the act was purely accidental, and how much of the act was negligence, etc.?
Judging black and white actions can be done by a human court. Judging intent (the thoughts of man), cannot be judged by a human court, only by G-d (who knows the thoughts of man).
Only G-d knows how much of the act was the fault of the person, and therefore G-d decides how much punishment the killer needs. G-d arranges it so that the murderer spends exactly as much time as he needs to to in the City of Refuge, in order to atone for his sin. If the killer was only marginally at fault, he would need to spend less time in the City of Refuge. If it was less of a pure accident, he might need to spend more time there.
As Rashi (Shemot 21:13) tells us:
...two people--- one killed accidentally and the other killed with premeditation but there were no witnesses who could testify [against them]. [Hence] the latter one was not executed and the former was not sent into exile. G-d then causes them to meet at the same inn. The one who killed with premeditation is sitting under a ladder and the one who killed accidentally goes up the ladder and falls upon the one who killed with premeditation, and kills him, and witnesses testify against him making him liable to be exiled. The result is that the one who killed accidentally is exiled and the one who killed with premeditation is killed.
Also Rashi (Bamidbar 35:25) brings two reasons:
As he causes the Divine Presence to reside in Israel, and lengthens their lives, while the murderer causes the Divine Presence to leave Israel, and shortens their lives, he (the murderer) is unworthy of being in the presence of the Kohein Gadol. (Thus, he must remain confined in exile until the passing of the Kohein Gadol. - Sifri, 20.)
Another interpretation: Because the Kohein Gadol should have prayed that this pitfall not occur in Israel during his lifetime. (Because of the kohein's fault in not praying, the murderer's exile ends with his death (Makos, 11a). Gur Aryeh objects to this intepretation, contending that Makos does not say this, but merely that, if the murderer should pray for the death of the kohein, his prayer might be effective because the kohein did not pray.)
Lord Sacks addressed this in his weekly message not long ago.
The Talmud explains that the Cohen Gadol bears some minute amount of responsibility; "as he should have begged for compassion." The simple explanation is that G-d gives people the free will and ability to do evil things, but this case concerns a mistake. Had the Cohen Gadol prayed more, perhaps G-d would have prevented the mistake from happening.
Though I've also heard it interpreted: "he should have begged the people to treat each other with mercy", causing an increased value of life throughout the population and therefore better safety practices, ergo less accidents.
An alternative explanation is given by Rambam in his Guide to the Perplexed: when a national tragedy occurs like the death of a Cohen Gadol, the people are united. So the hot-blooded relative of the victim (the "blood redeemer") will put aside his grudge and lose interest in killing the by-mistake ("accident" can imply zero culpability) murderer.
1. Question Restatement:
Why do accidental killers go free, only after the Kohen Gadol dies?
2. Answer, Unconditional Advocacy is Sufficient to Bring Life:
Obviously, this answer does not represent "Oral Law", but other Jewish traditions.
The role of a Priest is often misunderstood - they were only ever Judges of last resort, and their primary role was advocacy.
It is incredibly significant that God ordained advocacy and mercy to be the final word in judgment.
This is one of those truths that "either clicks" in you, or doesn't. But - it is all over Scripture, and all of Scripture can be interpreted in view of God's precept that unconditional advocacy is sufficient to bring life. (Eve's advocacy for, (and not against), Adam in the Garden; Moses' advocacy for Israel from the mountain; Rahab's advocacy for spies; Samson's advocacy for Israel; David's advocacy on the threshing floor; Job being restored - only after he advocated for his friends; and on and on and on). The opposite of this truth is all over Scripture too: God will judge according to our own judgments - and if we judge with condemnation, then no right subsists for us to plead for unconditional mercy.
The Kohen Gadol was a part of the judgment, and fully aware of their responsibilities to advocate for a killer - for the rest of their lives. This is always what it meant to be a priest: unconditional advocacy.
And "life" was the value of their unconditional advocacy.
I don't even know where to begin providing Scriptural support for this - because it is literally all over the place - and ingrained absolutely in all priests, and a constant reminder through every blood sacrifice offered. But, let me know - and I will update to address any question/objection.