Everything you'd want to know about assassination markets.

but how is this helping society be pretty swell for most people, and what is the one guy's job exactly?

Incentive to cooperate? A reduction in the necessity of war, which is by nature an inefficient use of resources? From the story:

The wise men of that city had devised the practice when it became apparent to them that the endless clashes of armies on battlefields led to no lasting conclusion, nor did they extirpate the roots of the conflicts. Rather, they merely wasted the blood and treasure of the people. It was clear to them that those rulers led their people into death and iniquity, while remaining untouched themselves, lounging in comfort and luxury amidst the most crushing defeat.

It was better that a few die before their time than the many. It was better that a little wealth go to the evil than much; better that conflicts be ended dishonorably once and for all, than fought honorably time and again; and better that peace be ill-bought than bought honestly at too high a price to be borne. So they thought.

Moving on.

(Can you not bet on the deaths of arbitrary people, only people it is bad to have around?

Nope, "badness" is determined by the market.

Is the one guy supposed to determine who it's bad to have around or something and only allow bets on those folks? How does he determine that, if so?)

The "merchant of death" diffuses the legal culpability associated with betting on the assassination market. The tension in the narrative comes from him feeling ever so slightly morally culpable for the assassinations, even though he only "causes" them indirectly. Again from the story:

Through judicious use of an intermediary (the merchant of death), the predictor could make his prediction, pay the fee, and collect the reward while remaining unknown to all save one.