Airing one’s dirty laundry is a daily event at Yahoo!, if by “laundry” you mean writing and by “airing” you mean publishing it to hundreds of millions of people around the globe.
Airing out one’s dirty laundry means exposing it to fresh air, generally to remove unwanted odors. Doing it literally means you actually do it. You don’t pretend to do it. You don’t metaphorically do it. You do it.
I guess that’s the advice from the writer for Yahoo! Shopping. Don’t expose your dirty clothes and wet towels to the sun and fresh air. Hide them or take them to your parents. Something stinks about that and I’m not referring to the laundry.