How are antioxidants useful as preservatives?

Answer 1

Antioxidants prevent oxidation usually by sacrifically becoming oxidized themselves. Generally there are two things you want to worry about in preservation: a) microorganisms such as bacteria and molds, for which antioxidants don't matter at all, and b) oxidative damage against which antioxidants protect.

If you want to see a really nice example, cut an apple in half. On one half rub a bit of a crushed vitamin c tablet (or a little bit of orange juice), and on the other half don't do anything. You'll see the untreated half will become brown much faster, due to the oxidative damage that the vitamin c is preventing in the other half.