If you make a cylinder with a strip of paper, there will be two edges, a top and bottom. If you make a Möbius band, these two edges will just be a single contiguous edge, which is what is meant by "boundary circle".
The easiest way to see what happens when you contract this circle is to draw a rectangle with arrows to indicate how two opposite edges will be glued together. Now squeeze the two sides that aren't being identified, until you have a circle with two marked points (though they are identified in the gluing), and two arrows on either side going in opposite directions.
So now it's clear: you get a disc, with its boundary glued along the antipodal map. In other words, you get the real projective plane.