I think I can help here.
The way flux works is that a loop will try to counter the change in the magnetic field. In this scenario, the loop is the one with the bar and resistor. As the bar slides, this is increase the area of loop, thus increasing the flux. The equation for flux is this flux = (area) x (magnetic field) and as you can see increasing area, increases flux. Because the loop doesn't like changes in the magnetic field felt (not sure how to describe this), it will counter this change by inducing its own magnetic field to counter the increase in flux. It does so by causing a current to run through the loop. Look at the magnetic field inside the loop and use the right and rule to see how the loop would generate its current and field. Since the magnetic field is pointing down, you want to curl your fingers upward with your right hand (the curl should be done within the loop). Now look at the direction of your thumb, this is the direction of the current. This should be in the counter clockwise direction. The loop will generate a this magnetic field via a counter clockwise current until there is no change in flux (i.e. when the bar stops moving).
I know that this is difficult to understand so I will also link you to a video which really helped me understand the concept.
