It sounds like there are quite a few things you can learn about the projective plane from this question. Let's first review the picture of the projective plane as a "completion" of the regular plane.
The line at infinity
In the regular plane, you have parallel lines. Consider the set $S$ of all lines parallel to a particular line. In projective geometry, no two distinct lines are parallel, so something needs to be done about this. The solution is that we introduce a new point (an "ideal point") which was previously not in the picture, and which lies on all lines in $S$. That is, it is their intersection point.
There are obviously many distinct collections like $S$, all pointing in different directions, so we'd have to add ideal points for those too. Once we have added an ideal point for every possible collection (actually they're called pencils of parallel lines) you finally have the property you want: every two distinct lines meets in exactly one point. The collection of ideal points is the "line at infinity" that the author is alluding to. It is "way out there" on the boundaries of the plane.
The thing to realize is that the line at infinity is just like any other line in the projective plane: every projective transformation maps lines to lines, and the line at infinity is just a particular one of those lines.
Let me now try to rephrase what the author said:
One can view ellipses, parabolas and hyperbolas as the image of a circle in the plane after a projective transformation of the plane. We stipulate that the line we have drawn is mapped onto the line at infinity (some line has to map there!)
If the line does not intersect our circle, then the line moves off to infinity and the plane is stretched a bit so that the circle becomes an ellipse.
If the line is tangent to the circle, then when it moves off to infinity, it takes the point of tangency off to the line at infinity "way out there." None of the other circle points are "way out there" though, so they all remain in the regular plane. The result: the circle is stretched out so that the points approaching the point of tangency stretch out across the entire plane trying to reach the point of tangency (which is now an ideal point, not visible on the regular plane.)
If the line is secant to the circle, then the line is going to take two points of the circle away with it to the line at infinity, out of view. The rest of the points remain "in the regular plane." Points on the perimeter of the circle now have to race off in two different directions to find the points of intersection with the line that went to infinity.
I hope this description helps you interpret that author's description.
I do not understand how the circle could ever be anything other than a circle...
Well, you are carrying some conceptions about Euclidean geometry into the projective picture, then. But many such conception don't apply. One big one is distance, which is not an invariant in projective geometry. There isn't really a metric, so even the notion of a circle (points all lying at some fixed distance from a single point) is not really defined.
Yes, in Euclidean geometry, a circle is what you think it is, and any Euclidean transformation will move circles to circles, e.g. like a translation, rotation or reflection of the plane. But in projective geometry, you have more transformations to work with. If you break away from orthogonal transformations and start using general linear transformations, you'll find that a circle is always mapped to an ellipse, and you can get between any two ellipses using an affine transformation. Projective transformations bring in parabolas and hyperbolas as images of ellipses in the same way.
I don't understand how I could get any of these shapes, when I am mapping through a line.
We are not mapping through a line, we are considering a line which is the preimage of the line at infinity for some projective transformation. The line in this example does nothing to determine the transformation we are using. In fact, many projective transformations are capable of mapping the given line to infinity.
To get a better grasp on this stuff, I recommend you read about doing projective transformations in terms of homogeneous coordinates of the projective plane. It is very similar to linear algebra. Specifically, model the real projective plane as homogenous coordinaes in $\mathbb R^3$, and projective transformations as $3\times 3$ real matrices acting on the homogeneous coordinates. You should be able to construct a transformation that sends the line drawn to the line at infinity, and when you check the three circles' images, you'll find they are as promised.