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:
Also Rashi (Bamidbar 35:25) brings two reasons:
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.