This story is pretty bizarre. But there are a few lessons to be learned here!
This woman told her Southwest flight crew that she had a life-threatening allergy to dogs-- there were two dogs on board. The woman didn't have an allergy certificate, or any kind of documentation about this alleged allergy with her.
According to Southwest's policies, if the crew is unsure whether or not they can safely get you to your destination, they will remove you from the flight. After this woman disclosed her allergy to flight crews, but could not produce any evidence, the flight crew asked her to leave the flight. The airline offered to rebook her on a flight the next day.
She refused, and eventually the pilot called for police. According to Bill Dumas, who recorded this video, the police spent a long time talking with the woman, and asking her to leave the flight before forcibly removing her.
So future reference, if you have a live-threatening allergy, you should always have the appropriate documentation with you.
And if police, flight attendants, pilots, and fellow passengers ask you to leave a plane, just do it.