What is the direction of current flowing through the primary coil and secondary coil of a transformer?
The current is flowing the way you think it should be, but the perspective is odd so it looks like the current is flowing the same way in both loops. It's clearer a bit later (few seconds) when they show the transformer core. 

The left inductor is wound opposite of the right inductor; the top leg of the left inductor is pictured below the bottom leg of the same inductor, so it looks like it's in line with the bottom leg of the right inductor.