Feet dry,
As temperature increases, Takeoff thrust decreases, resulting in lower Maximum Takeoff Weights.
If the Actual Takeoff Weight (ATOW) is less than maximum, thrust may be reduced to the point where the ATOW would be limiting at a higher temperature. This temperature (Flex for Airbus, or Assumed Temperature for Boeing) is applied to the engine computer, and the lower thrust produced meets all of the performance requirements, as though the aircraft was truly at Maximum Takeoff Weight. (It does slightly better, but that's another story).
The 2 big advantages are that (1) Engine stress is significantly reduced, thus increasing engine life and reducing the risk of engine failure, and (2) Maximum rated thrust for the true environmental conditions is still available if needed.