Indoor ice rinks are generally created by flooding a concrete pad with water. This pad contains pipes that circulate brine, the temperature of which can be reduced to below freezing. The brine is pumped through the pipes in the pad, and goes back to the refrigeration unit, where it is re-chilled and pumped out to circulate again. When water is put on the pad that has been made cold by the chilled brine, ice forms: an indoor rink. As long as the chilled brine circulates, the ice will remain. The arena temperature is generally comfortable for spectators–IME, about 65-70F/18-20C or so.
A Zamboni has two functions: to remove the snow created by people skating on the ice, and to lay down another level of water that (almost) instantly becomes ice. This way, the ice doesn’t wear down to the concrete.