Thats the KHF logo. The hand is critical for Hapkido technique. It defines our art in a way.
Grab your own wrist with all fingers grasping and twist around. Then do it extending the finger and and push down toward that finger and away. Its a bit tough to detail typing but it changes the technique, grab and pressure.
It makes a huge difference.
Attached is my KHF logo and school name below : )
Yes, the hand is known as "live hand"* as is a part of many Hapkido techniques (well, all the joint manipulations, obviously you don't punch like that and kicking is kinda up to you) Like dortiz says, it'a hard to explain in text but it increases the effectiveness of the mechanics of a lot of techniques. It also presents sorta a visual reminder because in a number of techniques, you can point the finger in the direction you want to go.
So it's not much more a symbol than the flying sidekick is a symbol in the middle of many Tae Kwon Do patches and such It's not an abstract but a direct representation of a characteristic or identifying part of the art
*It seems 'live hand' is also used to mean the fully splayed open hand as well.
I have always heard it referred to as the "Ki Finger". As described it shows the flow of Ki in the technique and also facilitates some techniques. Usually activating a pressure point or increasing leverage.