The avenger of blood would have been a relative of the person that was killed. This relative had the right to kill the person who killed their relative with no repercussions for the act. The same basis as eye for an eye and tooth for a tooth.
While the person who committed the murder was in the city of refuge, no one could kill him less they be held accountable for the death. If they killed the person in route to the city, it was ok. They could stand outside of the city walls and kill them before they made it through the gate and the relative was not to be punished.
If the person was found innocent, not a premeditated killing, they could remain in the city of refuge until they died of natural causes with no fear of attack from the avenger of blood. If they left the city walls, the avenger could kill them. While within the safety of the walls though, they were safe.
If the high priest of that city were to die, all who were under the protection of the high priest were allowed to leave the city as the death of the high priest was considered to atone for the wrongs of the people that had sought sanctuary.
Christians consider Jesus / Yeshua to be the High Priest. On the cross, He pronounced forgiveness. He has atoned for all who seek Him for sanctuary. Another typology if you want to see it.