The explanation behind this high content of antioxidants is probably that because the olive is a fruit that is exposed to the air, it has to defend itself from oxygen.  Vitamin E contained in olive oils, carotenoids and phenolic compounds are all antioxidants whose activity has been demonstrated in vitro and  in vivo. Phenols have countless biological properties, for instance hydroxytyrosol inhibits platelet aggregation and it is anti-inflammatory and oleuropein encourages the formation of nitric acid, which is a powerful vasodilator and exerts a strong anti-bacterial effect.

Concerning olive leaves, they contain the highest percentage of polyphenols and several reports have shown that olive leaf extract has the capacity to lower blood pressure and increase blood flow in arteries. The major constituent of Olea Europaea is Oleuropein (mentioned before), a bitter compound which "has been shown to be a potent antioxidant with antinflammatory properties".

Oxidation is a process that occurs continually inside the body, giving rise to the formation of free radicals. Free radicals do not cause severe damage thanks to the protection provided by antioxidants, which help to keep a balance up to a point. If the balance is spoiled, however, "oxidative stress" occurs, leading to deterioration of normal cell functions and even cell death.

 olives

 

 <<Free radical formation is associated with the normal metabolism of aerobic cells. The oxygen consumption leads to the generation of a series of oxygen free radicals. The interaction of these species of molecules of a lipidic nature produces new radicals which may interact  with biological system (...) Flavonoids, phenols have an important antioxidant activity towards these radicals>>

It is very important to prevent the oxidation of cells because chain reaction may damage cells. Moreover oxidative stress can be considered as either a cause or consequence of some diseases including Alzheimer's disease, Parkinson's disease, the pathologies caused by diabetes,rheumatoid arthritis, and neurodegeneration. 

Oxidative damage in DNA can cause cancer. It has been proposed that polymorphisms in these enzymes are associated with DNA damage and subsequently the individual's risk of cancer susceptibility.

 

 

Full study here: 

https://www.researchgate.net/profile/Obdulio_Benavente-Garcia/publication/284994504_Antioxidant_activity_of_phenolics_from_Olea_europaea_L_leaves/links/56616cd408ae4931cd59eb08.pdf

http://www.internationaloliveoil.org/estaticos/view/89-the-antioxidant-properties-of-the-olive-oil