Plasma
BM,
Yes, it is the plasma that burns (not the electricty). The plasma is the "4th state of matter" and is a very good conductor of electricity. The initial microscopic breakdown of air into ionised components allows a very rapid build up of heat therefore more ions therefore more conduction therefore more heat - hence a "power arc" develops in microseconds after the initial discharge (much faster than you can move your hand away...). A similar sequence applies with lightning strikes.
The very nature of this stuff is such that there is little analytical information available - it is all empirical/anecdotal. My guess that the reason power arcs track the skin surface is that the plasma is so hot and so well established that it is actually a lower resistance path than the wet body fluids. This would fit the instances of these observed surface burn injuries.