If the family was picked at random, a GG family is twice as likely to have resulted in us finding out one of the children is a girl as a BG or GB family (and a BB family would be ruled out by observation) so you end up with the intuitive but in this case also entirely correct 1/2 chance.

If the family was picked from only those that have at least one girl, a GG family is equally likely as a BG or GB family (and a BB family would be ruled out again but in this case by definition) you end up with the very unintuitive 1/3 chance the article describes.

So the initial filtering is necessary to create this "paradox" (it's actually not a "paradox" but a "problem", as others have mentioned). Without it, the intuitive answer is actually the correct answer.