The skin damage is related to the destructive activity of free oxygen related radicals produced by skin cells. Polyphenolic components of olive oil have been compared to traditional antioxidants, such as tocopherols, used by the cosmetic and pharmaceutical industry to prevent skin damage. Results show polyphenols as having the highest activity as radical scavengers.
There are references about the protective effect of olive oil in ancient Greek texts, a commonly used skin conditioner at the time.
A few small scientific studies support this. Warning: this does not imply that it is safe to apply olive oil then suntan. No matter what you put on your skin, sun exposure will eventually age and damage the skin. Avoiding the sun, wearing protective clothing or at least using the newer UVA and UVB sunblocks would be better.
What about eating olive oil to protect yourself from the sun? Eating a sauce made by simmering tomatoes in olive oil has been shown to protect the skin from sunburn, about the same amount as using a sunblock with SPF of 2 or 3. The thinking is that the lycopenes in the tomatoes are taken up by the oil .
If eating antioxidants protect you, why not just take vitamins and supplements which are antioxidants? Oddly, several studies show that while a diet rich in foods which contain antioxidants lowers risk of heart disease, just eating the supplements increase risk of heart disease and even skin cancer.