My first actual precise introduction to the geometric concept of infinity came from reading about inversive geometry as a kid.
An inversion about a circle of radius $r$ consists of moving every point in the following way: you draw the ray from the center of the circle through the point and let $d$ be the distance from the point to the center. You then move the point to the place on the ray where it is a distance $r^2 / d$ from the center.
Henceforth, I will call the center of the circle we are inverting about the "origin".
This geometric transformation has some pretty neat properties (and is closely related to linear fractional transformations of the complex plane): circles that don't go through the origin get transformed into circles. Circles that do go through the origin get transformed into lines. Lines get transformed into circles through the origin.
Of course, the transformation isn't defined at the center. So the book said that we'll add another point to the plane called "infinity", and an inversion swaps the point at infinity with the center of the circle.
This naturally fills in some edge conditions: e.g. when I said a line gets transformed into a circle, that's not quite true: the circle would actually be missing a point (the origin). But after adding the point at infinity (and decreeing that it lies on every line), the inversion of a line truly is a circle. And similarly, the inversion of a circle through the origin is well defined at all of its points and it becomes a line (complete with its point at infinity).
Through studying inversive geometry, it became pretty easy to intuit the idea of the point at infinity in terms of actually being a point; there is no need for any of these nebulous heuristics like trying to think about limiting processes or ill-defined quantities that somehow keep growing in size.
I'm not sure how useful this will be to you, though, since the geometric concept of infinity has very little in common with the infinite quantities that appear in set theory.