Thats a product of social norms and values. When a person does something that is outside of them, there are consequences. A lot of people don't like what he said, so they refuse to watch his show. If they see A&E as supportive of his views, they boycott A&E.
If A&E doesn't want to appear supportive of his views, they have to do something like pull his show. Otherwise they suffer from public backlash. It's a matter of personal choice acting with a collective force.
I didn't make the rules, but I do enforce them. I don't watch stuff that I find offensive or patronize companies that I don't agree with whenever possible.
It abides by the ethical paradigm that Good Business begets Good Business, and offending people in such a way is not considered good business.
He still has his practice of bounty hunting, just not a show on A&E. Whether or not his own personal practice suffers is up to his consumers.