I think you put your finger right on the reason: they wanted to use him as an example. The draconian fine would serve to warn others of what possibly awaits them if they do what he did.
I have no idea of what an appropriate fine would be. For him, maybe something on the order of $5,000 would dissuade him from doing it again—that sure would stop me if I were doing what he did. But, as you point out, that huge fine serves as a warning to others.
He was offered a $4,000 settlement and he turned it down, so they wanted to make an example of him to deter anyone else from making them go to trial. The damage award was set by the jury, not the judge, and the defendant came across very badly at trial-- he was forced to admit at trial that he had personally downloaded the songs when he had previously testified under oath, numerous times, that he hadn't. (At one point, he tried to blame his sister for downloading the songs using his computer; at trial, that was proven to be impossible.)