Antioxidants help to control the free radical damage by getting rid of the free radicals. They can donate an extra electron to the free radical, filling that hole and fixing the behavior. The free radical no longer goes around causing damage.
The skin does have it's own antioxidant system to help to counteract these free radicals, however over time the free radicals can continue to damage cells and their DNA. In addition, your body's antioxidants decrease with age, leaving the free radicals unchecked, and leaving your cells, cell membranes and DNA more vulnerable to damage. This results in loss of firmness, radiance, elasticity and can contribute over time to aging of the skin. Notice that I said over time. Because in theory you're experiencing this type of damage all the time, though more so after exposures to sun and such. But, the damage is cumulative and although you won't be able to see results right away, chances are you will see them in the long term. I think of this as similar to wearing sunscreen. You might not see a big difference now, but if you compare yourself to your friends that didn't do it in 10 years, you'll likely see a difference.