The way I read it, he had a 1st class ticket, but his kids were traveling economy. So he was trying to board them at the same time as first class, even though they were not.
There is only one class on Southwest: Cattle Class. When you check in prior to your flight, you are assigned a boarding group and number. Groups are A, B, and C from 1 through 60. A1 through A15 are reserved for Business Select and other special privileges (including frequent flier miles). Other than getting to board earlier and have a wider selection of seats, they are all the same.
According to Southwest's policy, people travelling together but with different boarding positions have the option to board together, provided the person higher up in line waits with the people further back. How this applies to families, I'm fuzzy on, but I would assume if you have a business select or other pass that allows boarding in the A1 through A15 group, it would make sense to have young children (say, under 10 years old) board with you. It seems like this is what the guy had done on several flights previously.
What the gate agent did was apply the boarding policy in the strictest possible terms, which IMO was an asshole move. But it was still technically according to policy. Did he get lucky, or was this particular agent just being overly strict? Could be either or. Pulling them off the flight for a tweet, however, was completely uncalled for. Threatening to call the police unless he deleted said tweet was harassment, plain and simple. Plus, how in the hell did she figure out who it was so quickly? Was she on twitter while she was supposed to be working, or did some corporate wonk call the gate?
I've been a customer of Southwest for a while, but how they handle this in the long run will determine wether or not I continue to be.