Why are some cables 'directional'?

And does the arrow point away from the source (away from the amp), to the speakers?
Yes, the arrow points in the direction of the signal flow. While there may be some disagreement that the actual wire may be directional, there is general agreement that the shielding employed in these cables requires a directional connection.
The shield is terminated only at one end. This may or may not have any effect, depending on the noise environment of the system.

I disagree with your arrow directionality (i.e. "points in the direction of signal flow"). Analysis Plus tells the user to point the arrows of their ICs toward the preamp/control amp. Probably like you say, has something to do with the shielding and grounding.

However, I do not believe that cables are directional re: signal only. As the current flowing through an audio cable is AC, by definition, it flows in both directions, not in one direction as in DC.
Sometimes, as in the shielded 300 ohm leadin from my FM antenna, the shield is left unterminated at both ends. What's best may depend on your particular equipment, and the environment it operates in. Try it both ways. One may be better, or there may be no difference. Or the whole one-end-terminated feature may be unnecessary.
Could be from the direction the wire was drawn through the die and formed the crystaline structure in the wire.
Some of the electrons are retarded and need visual cues as to which way to travel. Some of the better wires have "Do Not Enter" stamped on the wire side of the insulation so that the really stupid ones don't get confused.

Electronics like to travel on smooth and straight surfaces. So your system will sound better when you keep the cables as straight and as parallel to the floor as much as possible. That's why cable elevators are so important. They prevent wires from tangling and keeps the cables as flat as possible. Think about it, your car engine hates steep hills. Your amp will appreciate you smoothing out the elevation as well. And, try not to use any splitters. The electrons might get confused at the intersection.

With AC electricity, like music, electrons don't "travel" anywhere, but just jump back and forth in one place. Take it from me. Some of my best friends are electrons.