Specifically, the vitamins have some double bonds which can be rearranged when the hydrogen atoms leave. When a vitamin gives another molecule a hydrogen atom, the vitamin is said to have reduced the molecule. Reduction is the opposite of oxidation, which is what oxygen does and what is done to food molecules to get energy out of them. Vitamins are called free radical scavengers because of their ability to react with and thus get rid of free radicals. They are also called antioxidants because they reduce other molecules.
Peroxides react with DNA to cause mutations and react with membranes.
Peroxides also can react with molecules in the cell to form free radicals. Rats fed a high fat diet got less cancer if they were also given antioxidants. Cooking of fats turns them into peroxides. You can see this by the food getting brown. You can sometimes taste the peroxides as a bitter taste.