Why graph planarity is important
For some graph classes $C$, the question "is there a fast algorithm for deciding whether a graph $G$ belongs to class $C$?" is perhaps only of theoretical curiosity. But it can also be argued otherwise: suppose a problem you care about is hard in general, but efficiently solvable for graphs of class $C$. Wouldn't it be nice if you could quickly test if you can use a fast algorithm, instead of always using a slow brute-force, for example?

In general, it is not hard to imagine some graphs arising from applications are naturally planar, like road networks, printed electric circuits, railways, or chemical molecules. For more concreteness, it seems like a good set of keywords is "planar graph applications". One of my first hits was [1] from the domain of computer vision. 



[1] Schmidt, Frank R., Eno Toppe, and Daniel Cremers. "Efficient planar graph cuts with applications in computer vision." Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on. IEEE, 2009.