It's a very specific concept from the United States Constitution that relates to the government forcing you to refrain from saying things that you want to say. Freedom of speech is the kind of thing that comes up when a government agency puts warning labels on record albums, news media are prevented from reporting on government activities, or butthurt lawyers (cough cough Charles Carreon) try to use the courts to prevent people from criticizing their clients.
The blogger in question felt that the word "badassity" was an essential element of his brand, and that to change his header would dilute his brand in some way. He made a calculated decision that appeasing Chase would negatively affect his brand, perhaps even resulting in a reduction to his overall income in the long run, either directly or indirectly. A post like this one reinforces his brand as a blog with "integrity," and an attitude, and is sure to help offset whatever immediate loss he sees from Chase withdrawing their support.
While he's clearly a skilled and savvy blogger, I really don't see this as a free speech issue. A more accurate title would be "I Just Gave Up $4000 Per Month to Preserve My Brand Integrity."