I think the point is that the more compression, the higher the pressure in the combustion chamber, and the more the gas expands in the turbine section. To me the real question is why is the energy gained from the extra expansion is greater than the energy used to compress the gas further in the first place.
The only thing I can figure out is that there's effectively less gas in the compressor section (because it hasn't yet been heated in the combustion chamber), and compression therefore take less energy than you get back from expansion.
Imagine climbing a mountain with a 10 pound weight, then letting it go to roll back down. If you had a magic weight that increased to 20 pounds when you let it go, the weight would release more energy rolling down the mountain than you put into it climbing up. And the higher you climbed before you released it, the greater the energy difference would be.
Heating the fuel-air mixture in the combustion chamber is a little like increasing the weight from 10 pounds to 20, except it's due to the heat energy from combustion, not magic. But it still means you get back more energy from the turbine than you put in with the compressor.
At least that's my (doubtless somewhat confused) story.