Sorry I should have specified which breather tube. 
The big breather tube coming from the valve cover goes over the carb and air box.
It the runs down the back side of the engine to where???
It doesn't seem to fit between the swing arm and engine and there seems to be no guide or holder.
How does it exit out the bottom without blowing all over the swing arm and engine?

it just hangs there. Many will afix something to hold it still. Racers are often required to dummy up a catch can to run these open air breathers into.
As far as "all over the swingarm", the only thing coming out of there should be vapor. If anything accumulates you've got an engine problem. In constant high RPM racing environment you could get some drops to accumulate over time.
I think there is a metal loop attached to one of the 3 large case bolts at the rear of the engine to capture the ends of these tubes, breather, carb, and battery. It is often discarded (forgotten) in engine rebuilds.
The battery tube should go down between the engine and swingarm to the lowest point possible otherwise its fumes will degrade the finish on the swingarm and perhaps damage the chain.