Unlike most bipeds, who walk by shifting their center of gravity towards the direction they want to move, the Otto DIY robot doesn't seem to have any clear shift in center of gravity. This is probably what makes its "walk" so confusing to analyze. While it does not shift its center of gravity in the direction it plans on walking, it shifts its center of gravity nonetheless.
If you analyze the Otto very closely you'll see that right before it starts its "ambulation", it flexes one foot up ("toes" up) and the other foot down. The foot that is flexed up does not affect the center of gravity of the robot. However, the foot that is flexed down lifts the robot up on that side causing the robot's center of gravity to be shifted towards the flexed down foot. We should note that the surface area of contact between ground and foot for both feet is reduced when the feet are pointed. This is important as the foot pointed up now has a normal force from the ground almost colinear to the "leg turn" (as you've labeled it) motor.
So, with the center of gravity now "leaning" towards the foot up leg, the robot's body will rotate over this leg when the motor rotates. This process is then repeated with the other leg to accomplish what is closer to waddling than walking.
Of course, the actual analysis of such a robot would be more complicated than this. For instance, if you implement a similar design yourself you may notice that the foot up leg twists about the leg turn motor instead of the robot's body twisting about the leg. In this case you would need to look at the actual dynamics of the robot to figure out what torques (if any) will cause the desired motion.