The origin of The Illusive Man's eyes is told in the one of the Mass Effect comic books, titled Evolution, the full plot summary of which you can read here on the Mass Effect wikia. It's a 4-issue miniseries, so it's a bit long to completely paste here.
In it, you learn the origins of The Illusive Man and who he was before becoming the leader of Cerberus. I'd highly recommend reading the issues yourself, as it's a nice bit of background information, but I'll write up a quick summary about his eyes.
During the First Contact War, he was part of a mercenary group working for General Williams (Ashley's grandfather). He was exposed to a Prothean relic on Shanxi. Despite the fact that others who were exposed to when it activated ended up basically turning into Husks and losing their higher cognitive abilities, he did not change. However, his eyes were permanently changed and he is able to understand the language that the Husk-like beings now speak. You get to see the more human side of The Illusive Man, what happened to him to engender his pro-humanity stance, and why he believes in it so strongly.
It does not further specify in the comic book series nor in the Mass Effect novels what exactly the impact of this change was, and it is currently part of the mystery of The Illusive Man. The fact that it was a Prothean Relic however, may change your interpretations of his actions and motives.