The term is usually used in reference to shots that miss the green. Let's say the flagstick is place on the right-hand side of the green. A golfer whose approach shot misses the green to the right is on the short side, and might be said to have "short sided" himself.
It's called the short side because there is less green (the green is shorter, so to speak) for the golfer to work with on that side. A chip shot from the right, to a flagstick that is placed close to the right-hand side of the green, has a much greater chance of running well past the hole.
Television announcers often refer to this by saying the golfer "doesn't have much green to work with."
If the hole is cut in the back part of the green, then a golfer missing long is on the short side, and so on.
