Earlier quoted context omitted.
He was in a position to hire people, and expressing views that clearly indicated bias towards certain protected classes. That alone would be cause for concern. I don't know that I"d advocate his firing, but I'd certainly question his authority to hire & fire staff based on the things he's said.
This generalizes further than just antifeminism. Christians and atheists are usually in positions to hire people. I guess we should just fire everybody who ever believed in anything, and then we'd have a perfect world, right?
It's quite simple (and already explained by rit). He posted numerous messages disparaging various groups - blacks, women, gays. He is responsible for hiring within the company. This places Business Insider in an utterly compromised position and in danger of lawsuits.