Something I tried which helped me to understand this very phenomenon:

Take a piece of string and a globe and drag the string between two points on the globe. When you pull the string taught it will naturally follow the shortest path which, counter-intuitive to us who grew up looking at flat maps, actually curves. I was surprised to find that the shortest path between Atlanta and Tokyo goes over Alaska.

Here's an example of something similar using a flat map:

http://maps.forum.nu/gm_flight_path.html