Context: “People like you” apparently butt out and pay their own bills. To those who appreciate irony, people like them are a four-legged answer to Christmas. I see no reason to take them seriously on any level, except as family — and that mostly consists of leaving unsaid 99 percent of the retorts that pop into your head.
This is a good example of context providing meaning. Without knowing that the original sentence is the response to a question soliciting advice, it is impossible to get why it works as a sentence.
Ms. Hax is responding to a question about what the person's next move should be. "I'm partial to falling off one's chair, laughing," meaning "My best advice is to find their behavior ridiculous." (To parse it out: "I'm partial to" = I'm in favor of; "falling off one's chair, laughing" = your relatives' behavior is too outrageous to be taken seriously.)
This is a response to what comes before it. Providing us with the sentences after it is not really the context.
The question is: What should you do when people who make you pay for things are critical of the fact that you work hard for your money. Her opinion is that you should laugh at them so hard that you fall out of your chair.