When you eat or exercise the body undergoes changes, not all of them good. One change is oxidation, where electrons are lost as part of a chemical reaction. Electrons normally spin in pairs around the center of an atom. Lose one and the atom, molecule or ion which loses it becomes a free radical. It tries to make up the missing electron by steaing one from its neighbour. This leads to a chain reaction that can result in DNA damage and ultimately cancer.
The body has several defence mechanisms to recover from oxidative damage. One of them is to make up the missing electrons from foods or vitamins (like vitamins C and E) that have the capacity to assist. An important note, when taken in together, antioxidants are multiplicative in their beneficial effect on the body rather than additive.
Some antioxidants are water based and some fat based. This means they each work with a different part of the body.
I did a search in Google for the antioxidant potential of the ingredients I use in my food bars and powders and was stoked to learn that when I did the search every single ingredient had antioxidant properties.