What are the antioxidants actually doing... and to which chemicals in the developer is it doing it to? (just the developing agent itself?)
An antioxidant is a molecule that counteracts or reduces oxidation reactions. Technically it applies to the oxidative effects of oxygen, it is commonly applied to any molecules that protect from any free radical (something with an unpaired electron).
Free radicals can be formed by the rupture of a bond in a stable molecule with the production of two fragments, each with an unpaired electron. These unpaired electrons are usually highly reactive, so radicals are likely to take part in chemical reactions.
Free radicals can be formed from atoms like fluorine and chlorine, inorganic ions like hydroxyl OH-, NO2-, or ClO3-, or organic like CH3, CH3CH2, and C6H6−.
Biologically, antioxidants include beta-carotene, lycopene, vitamins A, C, and E, and certain enzymes.
Antioxidants, like BHT (butylated hydroxytoluene) are also commonly added to food products like vegetable oils and prepared foods to prevent or delay their deterioration from the action of air. BHT is also used in fuels and other commercial applications.
Sulfite, as PE mentioned, is probably the most common photographic antioxidant. And oxygen from air is probably the primary oxidative concern in photographic applications.