Oxidation is a chemical reaction in our bodies that causes the degeneration of cells. The main byproduct of oxidation is free radicals. In turn, these radicals can start chain reactions. When the chain reaction occurs in a cell, it can cause damage or death to the cell. Antioxidants terminate these chain reactions by removing free radical intermediates, and inhibit other oxidation reactions.
In short, antioxidants reduce oxidation which reduces free radicals which reduces the damage and death of cells within the body.
With some exceptions, these unpaired electrons make free radicals highly chemically reactive towards other substances, or even towards themselves: their molecules will often spontaneously dimerize or polymerize if they come in contact with each other.
In short, free radicals reactive with other substances and degenerate cells within our body.