In basic terms a 'Flex' Takeoff is a takeoff that uses less than full thrust (which is known as a rated takeoff) It is usually used whenever the aircraft is below it's max takeoff weight to reduce the thrust to a level where it will be less then full thrust but still meet all the performance requirments even in the event of an engine failure at or after V1. The benifits of this include increasing engine life and reliability largely as a result of having a lower EGT for takeoff.