"Brownbacking is unprotected anal sex, or anal sex without condoms. The term should be understood to refer to risky unprotected anal with multiple and often anonymous partners and a self-identified "brownbacker"—someone with that bumper sticker on his car—is someone who seeks unprotected anal sex with multiple and often anonymous partners. Brownbackers prefer that term because it sounds sexy and healthy and glamorous and it lends itself to brownback-glamorizing bullshit. ... It associates shitty behavior and shitty choices with actual shit."
When a public official or his handlers demand, either directly or through third parties, that an average citizen apologize or beg forgiveness for having have hurt the official's feelings, and the everage citizen's defiance goes viral and forces the official in question into a humiliating climbdown in which the official winds up apologizing to the citizen instead.
After the mayor demanded an apology from a high school student for an allegedly disrespectul tweet, he got well and truly brownbacked by his angry constituents and had to say "I'm sorry."