Well..actually it isn't to dangerous, but it's a case of better be safe than sorry.
The high voltage lines can exceed 750,000 volts and that amount of voltage can actually jump across air if a ground is near by (think of stun guns where even 50,000 V can jump air across the contacts)
If you look at high tension lines carefully you will see insulators at least 5 ft long at every tower because the electricity could jump to the steel tower and short out if they didn't use those big insulators to route the electricity around the tower supports.
Air is a pretty good insulator which is why those lines are so high in the air and don't lose much electricity to the air.
So when you fly close to the high voltage lines and if your kite was to contact the high voltage the thought is the kite string might conduct the high voltage better than air and the electricity will take the shortest path with the least resistance to get to the ground.
But cotton string is a pretty good insulator in itself. But sometimes electricity is a funny animal and it could conceivably ride down the string along the outside surface and go thru your body on it's way to the ground. Because even though cotton string and nylon are good insulators that doesn't mean there might not be humidity in the string line which would also lower the resistance. The electricity doesn't actually go thru the string on it's way down to you, it rides on the outside.
Actually the best comparison is lightning. Trees are pretty good insulators but in a storm the lightning takes a shorter path to the ground by going thru the tree than just striking the ground. And you probably have seen what lightning can do to a tree. But the lightning does it the same way. The lightning rides along the outside surface of the tree in it's path to the ground, usually stripping off the bark of the tree and sometimes splitting it.
So in a nutshell...the high voltage line is bad for kite flying because potentially the electricity can follow the outside of the kite string down to you and the ground