Can someone please help me figure out this exercise?

This whole method appears needlessly confusing.  The field APPLIED is from South ( your side of the page) to North ( into the page).

Using the right hand rule on the direction of current the field CAUSED by the wire is OUT of the page in the area to the right of the wire.  ie the area on your side of the loop is the north face OF THE LOOP.

So there is a magnetic field entering the north face of the loop.  Reducing the area of the loop by moving the wire to the right reduces the amount of flux entering that north face.  The rule to which they refer says that the coil would move to reduce the amount of flux entering the North face. ie to the right.

Now I would just have used the right hand rule ( fingers in direction of field applied, thumb in direction of current, palm points in direction of force) But HEY why make things easy if we can find a more complex way of thinking.

To get the north face OF the loop picture a circle where you are viewing the left hand wire of that circle.  The current is moving DOWN the page.

Now get your right hand and put the thumb along the wire.  Wrap your fingers around the wire.  On the RIGHT of the wire ( inside the loop ) which way do your fingers point?  Out of the page is the direction.  So the coil PUSHES the field out of the page causing that part to become a North Magnetic Pole.  Ie we have the situation where it is being repelled by the applied field.  It will tend to do something to reduce the flux passing through it. 

The circle lies in the plane of the page.  The right hand rule then tells you that the north face is the face of the page facing you.   The magnetic flux is entering the north face of the loop.  Since F = iL x B where L and B are vectors, there will be a force pushing the loop to the right on the segment shown, and to the right on the segment on the other end of the diameter of the segment shown.   you can pretty much see the forces act radially inward so the circle is being squeezed to a smaller diameter hence area.  Since flux = magenta filed times area, the reduced area will minimize the flux