Wiktionary says it is either likely a pseudo-condensed pronunciation of kilometer or onomatopoeic of the sound of a military odometer.
Though kilometers are not commonly used to measure distance in the USA, “klick” is commonly used by the US military, which uses the metric system almost exclusively in order to facilitate communication with allied forces.
Here’s an example of the word being used in the film Apocalypse Now:
We’re going up river about 75 klicks above the Do Lung bridge.
Best Answer
I was in the military, but not Viet-Nam era. From my own judgment, I don't think it has anything to do with odometers. I think it's a dark humor reference.
When sighting in an M-16, the sights are adjusted in clicks. Looking downrange at a calibrated 100 yard target, one click moves the projectile/impact one inch across the face of the target. (left/right/up/down)
In the same way, when looking at a topo map, one "klick" is one full increment line on the map.
So viewed from the sky, (the point of view of either God or an enemy bomber, however you wish to imagine it) when we go one kilometer, we've traveled one click across the face of the map. The unspoken implication is that in the end we all might just be points on someone else's target...
Simply my opinion, but that's how it struck me the first time I heard it.