I do not know the man, never met him, but I will give it a go and answer your question.
Timing is the answer. It was just bad timing.
Brohm got his first big job with Purdue, and after a huge one year improvement with the team, he was on his way of building his reputation as a exciting, aggressive young coach. Purdue fans were excited and recruiting was upgraded.
Then all of a sudden, his hometown school where he played had to fire Bobby Petrino in the middle of the season. Brohm's world got more complicated as he knew he was the man who would get the first offer.
Now a self centered, selfish coach would have dropped Purdue with little to no regard of what that would do to the team. Maybe Jeff Brohm is not that kind of coach? Maybe he believes in loyalty, especially when he has the entire Purdue fanbase begging him to stay?
I think we all know Brohm wanted to ultimately get the UofL gig, but he had to weigh everything that could come from a decision to bolt to Louisville. I think the guilt of letting so many people down just one year into being the head coach outweighed his dream of being the head man at his Alma mater.
He didn't want to start his coaching career like the coach he would be replacing. There is plenty of time to get the Louisville job but only one time to start a career with integrity.
So he decided to stay at Purdue and trust that he didn't burn any bridges with UofL. I think he's got nothing to worry about because if Satterfield decides to move on or is told to leave. Jeff Brohm would be more than ready to accept.