An Arbiter fields two primary purposes:
For one, he is designated as the one person allowed to pass final judgement on his fellow human beings when they're at the mercy of the Legendary Council. The Sehto Legendaries discuss decisions largely amongst themselves, but they will look to the Arbiter for the final call. Since this privilege is granted by the Legendaries, it can, of course, equally be revoked at any time, effectively nilling its power in theory, though in practise the Legendaries are quite good at taking advice - from their current Arbiter, Jagdish Tsukinaka, at least.
For two, rather unlike what the term might imply, the Arbiter is responsible for the execution of the will of the legendaries once a consensus between him and the Legendary Council is attained.
Formally, the position of Arbiter began as a pure advisor to the Council. He had no power to fell decisions about the sentence of the accused, but he was granted the ability to veto these decisions, which was an indirect way of having much the same influence - if the Arbiter wished, he could veto all decisions but the one precisely conforming to his expectations, after all. That's also why this cornerstone's wandered over the decades.
Furthermore, the Arbiter was - by definition - on the side of the accused, albeit without any checks in place to assure that. (It's still important to the Council that the Arbiter is at least not maliciously inclined toward the accused.)
The combination of formal alignment and the ability to veto any decision formed the axioms for the third purpose - the execution of the sentence. The Council deemed it important that it was executed by someone on the victim's side, rather than someone eaten up by some desire for vengeance.
While a strong supporter of principle of an Arbiter, Jagdish Tsukinaka only reluctantly fills the role himself, because the associated chore of sentence execution is not entirely to his liking; while he fully agrees the victims of the Council deserve their (arbitrated) fates, emotionally, he would rather delegate the task of its execution to someone else. Rationally, he deems himself the best candidate, as it is precisely his lack of sadism that makes his actions so meticulously and precisely applied.