Well, upon a closer look, one notices that the chessboard coloring is not necessary for the problem statement. It's kind of a hint actually as you could equally well just consider a blank 8x8 board and realize that this coloring arugment works. I just feel the problem is unreasonably difficult that way.
The coloring is kind of additional structure that is applied on the object you are working with. And I think this idea of "applying structure" is a very generic. You can solve similar combinatorial arrangement problems that way, but it goes beyond that.
I think that a nice, classic (and significantly more advanced) example is showing that plane and punctured plane (a plane with one missing point) are topologically different. The fundamental (homotopy) groups of these spaces are different, and hence the spaces cannot be continuously deformed to each other.
Somehow the spirit is the same, I feel. In this topology proof it's not a grid you are working with, but a topological space. And the structure you apply is not a coloring, but something quite abstract (a homotopy group). The idea in both cases is similar, though: You apply structure and this structure reveals something that's not easy to see directly.
The magic part is figuring out the structure that produces the data you need.