I'm a little used to the trick but it'd be nice to give some deeper motivation of why the trick works in the first place and why this particular representation.
I guess the important part is clearing denominators turns problems with division into linear problems.
There are many similar results, but this is one of the most easy to understand.
The projective method extends an N-dimensional space with a set of points at infinity that forms an (N-1)-dimensional space.
That means that a projective straight line has 1 point at infinity, a projective plane has 1 straight line at infinity, a 3-dimensional projective space has a plane at infinity and so on.
The other 2 methods are exemplified by the extended real number line and the extended complex plane.
The extended real number line has 2 points at infinity, not 1 point like a projective line. This method is seldom used in higher dimensions, but nonetheless it works in any dimension, i.e. a plane can be completed with a circle at infinity, a 3-dimensional space can be completed with a sphere at infinity, and so on.
The extended complex plane has only 1 point at infinity, not a straight line like a projective plane. This method can also be used for spaces with any number of dimensions, which are always completed with a single point at infinity, regardless of the number of dimensions. This method is used in the so-called inversive geometry (which studies invariants to transformations that preserve angles in spaces with 3 or more dimensions).
While the other 2 methods of handling infinities are also useful in certain applications, the projective method of handling infinities is by far the most beautiful in my opinion, because it allows the unification of many concepts, simplifying a lot their handling. When I first learned projective geometry when young, it was quite a revelation. I consider only very few parts of mathematics and of sciences in general as beautiful as projective geometry.
When thinking about a problem, there is absolutely no need to think about the hack of homogeneous coordinates. That hack corresponds with a particular model of a projective space, where an N-dimensional projective space is mapped to a hypersphere embedded in an N+1 dimensional linear space, e.g. a projective plane is mapped to a sphere (with double cover).
When you make computations, homogeneous coordinates are the easiest, but when you think qualitatively it is easy to think directly in a projective plane or in a 3-dimensional projective space, without using any extra dimension. E.g. in the latter you have an extra plane at infinity. Each point in that plane corresponds with a straight line in the 3D space, and also with any other straight line parallel to it. Each straight line at infinity corresponds with a plane in the 3D space, and also with all planes parallel to it. Each circle at infinity corresponds with a conical surface in the 3D space, and also with all other conical surfaces that have the same vertex angle and the axes parallel between themselves (and also with all hyperboloids with 1 or 2 sheets that approach asymptotically those conic surfaces towards infinity). A polygon at infinity corresponds with a pyramidal surface and with all other pyramidal surfaces with parallel axes and equal angles. And so on, any geometric figure at infinity will correspond with some surfaces or curves in the 3D space, which end at infinity into that geometric figure. If you understand which is the figure at infinity in which a surface of 3D space ends, you can understand in what kind of surface the original surface can be transformed, when the figure from infinity is brought to a finite distance by a projective transformation, and another section of the surface goes to infinity.
For a 3D space, it is slightly more difficult to visualize a flat plane at infinity, instead of visualizing a single point at infinity or an enclosing sphere at infinity, as in the other 2 methods of handling infinities, but the rewards are greater, so the effort of understanding the projective method is worthwhile.