I know you can just read page 12, section 7.3 to find out that it is a reward for becoming the controlling faction. Not knowing a lot about this point in history, I take it as you gain the support of those who where on-the-fence about the debated issues and since you became the controlling faction, you simply won them over. Now you have more power to accomplish your plans to greatness.
An interesting theory, but not what I was simulating. Both sides had the equivalent of a Military Council. In the case of Athens it was the Strategos of 10 elected generals. The Spartans had the kings and ephors who did the equivalent.
If you are the Controlling faction you in essence are gaining strategos leverage over and above what was achieved in the political arena because you are the head of the military.