This actually helped. I always assumed that the goats/prize were randomly placed, Monty's Door selection was random, and the contestant chose a random door. In which case I could never fathom why you'd need to switch. Since as the author says, there's no new information about the remaining two doors, and switching or not switching remain random.

I still feel like I don't full grasp the actual problem/solution though. Even with this new understanding. I will read the linked article by Jeffrey S. Rosenthal though, and hopefully that will fill in the last of the blanks.