People work at a company for a variety of reasons. Some want or need the money. Some really believe in the company's mission. Some like the people they'll be working with. Some want to develop skills for their next move. Some want the brand name or status. Over time, the company will accrete people of all sorts of different motivations.
What a buy-out does is take money out of the equation. It makes it so that everyone who is there for financial reasons has a financial reason to leave. This can be logical if the company has little to offer financially, as in, for example, a company that's in bad shape during the financial crisis. The only employees who're left are those who are not in it for financial reasons - the true believers in the company, those who like their coworkers, and those who want to gain skills. Those people will be a lot happier when they don't have to listen about their financially-motivated colleagues grousing about how the company is going under, and they'll be more productive when they're happier.
The CEO's fatal mistake was that he forgot that his board is financially motivated. From their perspective, he's acting insane, because why would anyone stay at a company for reasons other than money? And they're his bosses, so they can have him removed. For this to have worked, he probably should've taken the company private, or at least packed the board with people who bought into his vision for what kind of company it should be.
The board's fatal mistake was in removing the CEO. By offering buyouts, he had just aligned the company in one direction. By removing him, they suddenly took the company in a different direction - one that they had less than zero capital in, because the CEO's previous actions had alienated everyone who might've helped them.
In both cases, the failure was in understanding that people are different and may have different motivations, and that a company survives based on how well it can hold together people who all have their own goals and worldviews and motivations.