Antioxidants in foods help protect your brain from all the tiny amounts of damage that happen constantly just because you are alive, living in today's world and vulnerable to damage done by changes in blood flow, illness and accidents. And it seems different ones help in different ways.
But you don't have to waste time and money trying to work out which ones to have - all you really need is to include as many different colours in every meal as you can because then you will get a wide selection of different types. And eat them in your usual foods and you'll also get them in the way they were intended.
For a more detailed explanation and food suggestions, Chapter 2 in Eat To Cheat Ageing is all about your brain.