Generally, a scale will measure the normal force it supplies to the object resting on it. In the special case where the scale is stationary (as it appears in your picture), this is equal to $mg$, or the weight of the object.
If the system is accelerating, the normal force (and thus the reading of the scale) will increase or decrease appropriately. However, this normal force is no longer equal to the weight.
The question in the problem is about the minimum speed at which the passengers will not fall out. Therefore, you may assume that in the upper point the normal force is zero (otherwise you can decrease the speed without the passengers falling out). Or, more precisely, it is possible to show that if the passengers do not fall in the upper point of the circle, they do not fall in other points of the circle, so, to find the minimum speed you can require that the normal force is zero at the upper point of the circle.
As for the normal force going in the direction of the weight... I cannot confidently say that this statement is wrong, because there are actually two very different definitions of weight (https://en.wikipedia.org/wiki/Weight): 1) "the weight of an object is usually taken to be the force on the object due to gravity" 2) "There is also a rival tradition within Newtonian physics and engineering which sees weight as that which is measured when one uses scales. There the weight is a measure of the magnitude of the reaction force exerted on a body. Typically, in measuring an object's weight, the object is placed on scales at rest with respect to the earth, but the definition can be extended to other states of motion. Thus, in a state of free fall, the weight would be zero. In this second sense of weight, terrestrial objects can be weightless. Ignoring air resistance, the famous apple falling from the tree, on its way to meet the ground near Isaac Newton, is weightless."
Under the second definition, the normal force is indeed going in the direction of the weight, but the formula in your question for the summation of forces is correct under the first definition:-) So you may wish to find out (or tell us:-) ) which definition your teacher uses.
Best Answer
I'll give an answer based on intuition (which illustrates the physics).
The scale only measures the portion of the weight that is applied to the scale.
Imagine you tie a string to a weight and place the weight on the scale. If you pull up slightly on the string the scale will read less, even though the weight of the object has not changed. The weight is counteracted by both the normal force from the scale, and the tension you are applying with the string. But the scale only "sees" the normal force.