In this quote, Gene is pondering the events that occurred during those crucial years at Devon school, and pairing it with his more mature knowledge of war and people that he has gained throughout his lifetime. Gene realizes that he, in a symbolic sort of way, had declared war on Finny. He ignorantly concluded--because of his own insecurities and jealousy of Finny--that Finny was trying to bring him down and distract him so that he wouldn't do well in his studies. He assumed that Finny always had people playing games simply so that Gene wouldn't do well in school. Finny struggled in school himself, so Gene concluded that Finny was jealous. It was that erroneous projection of his own jealousy that led to his reckless actions at the tree. He let his jealousy and ignorance guide his actions, and Finny ended up with a broken leg. That event eventually leads to Finny's death. So, Gene's ignorance of Finny's pure intentions and true friendship was what, in the end, led to Finny's death.
Compare that description to the quote you related. "War is made by something ignorant in the human heart." Gene's heart was ignorant--his own jealously led him to conclude that Finny was vindictive, when he was not. And that jealously led him to "declare war" on Finny in the tree. Finny was a casualty of that ignorance. Gene takes that very personal experience and applies it to war on a very large scale. He concludes that wars are started out of that sort of ignorance, not the larger, more sweeping evils that many people have often attributed it to.
I hope that those thoughts help a bit; good luck!