Just for clarification:
most engines(i know anyway) are flat rated.
A flat rate is quite simply a way of protecting the engine from surpassing it's limits when temperatures drop.
f.e.
when we do power setting calculations we refer to a form of maximum thrust, call f.e. N1ref
engine flat rated at 23°C will produce it's maximum thrust at it's maximum RPM(say N1ref=95%N1) at 23°C.
Going into a lower temperature, say 20°C, you enter the purely structural/thrust-limited section and you would see the resulting required RPM is lower, say N1ref=92%N1.
If you go into a higher temperature, say 30°C you enter the engine's temperature limited zone. this is where you meet your EGT/TGT/ITT whatever you call it - limits. again, result is a lower N1ref=92%
Flexing is making the engine believe it's hotter outside than it actually is.
Depending on you runway and conditions you can calculate a maximum ambient temperature for which running your engines at N1ref would give you still sufficient margin on your takeoff/climb performance required. (This is usually done by the company's engineering dept and presented to you in a performance chart specific for 'a runway/intersection/flap setting/runway condition' and contains values to correct for anti icing, bleed selections and wind.
So if it were really 'that' flex temperature outside, we'd be very sweatty and selecting our max thrust N1ref, would be able to takeoff from that runway.
So now we do this with a temperature which is actually lower than that flex temperature, giving us an additional margin on performance. And most importantly, we will have a lower N1ref setting sparing our engine and increasing its life thus reducing company costs.
It's important to know aswell that when your flex temp for a given rwy/condition/flapsetting is lower than the flat rate of your engine, you should not perform flex takeoff.