Two good ways to represent plane curves are parametric form:
(x(t), y(t))
or implicit form:
f(x, y) = 0
Parametric form is naturally associated with one point of view of what a plane curve is: the set of points traced out as a parameter is swept over its domain. Implicit form is naturally associated with a different point of view: the set of points that satisfy a certain relation.
Functions of the form
y = f(x)
can be easily re-expressed in parametric form:
(t, f(t))
or implicit form
y - f(x) = 0
so both the parametric viewpoint and the implicit viewpoint are equally valid and useful ways of understanding the graph of a function. You could rephrase the author's insight as saying that he had always understood graphs of functions parametrically, but later learned to also understand them implicitly.
Depending on the application, it may be more convenient to have a parametric representation of a curve, or an implicit representation of a curve. For example, it's easy to find a point on a parametric curve, but hard to test if a point is on a parametric curve; on the contrary, it is hard to find a point on an implicit curve, and easy to test if a point is on an implicit curve. If your curve is the graph of a function, it is easy to convert back and forth between these forms, but in general, converting from one form to the other may be quite hard.
For me, the relationship between implicit and parametric representations is a piece of math that I didn't really learn until long after I thought I had already learned math.