if you take a 2d piece of paper, draw the numbers 1 to 5 horizontally, then look at the distance between 1 and 5, or 1 and 3, they vary, now roll it into as cylinder, it's a manifold, and now the distance between 1 and 5 and 1 and 1 and 3 are the same, no matter how many numbers you had written on the paper, they would be connected by the shortest path, something like a helix I believe, the helix being the geodesic, now, increase the dimension, but keep rolling it up into that 'cylinder' and using a new geodesic to connect the shortest path, this is my layman's take on it, if you are #math please jump in and put me right!