The attached PDF shows the basic difference, in forces, between the normal cut and the climb cut.
In the normal cut, the router bit is drawn into workpiece by the force developed by the cutting edges on the workpiece. On the router table (as shown), the workpiece is drawn towards the fence. Also, the cutter applies a force on the workpiece in the opposite direction than the feed direction.
In the climb cut, the router bit tries to "climb" out of the workpiece. On the router table, the workpiece is being forced away from the fence. Also, the cutter applies a force in the direction of the feed direction.
This is what creates the danger for the climb cut. The bit moves the bit in the direction of the feed direction -- usually this means the worker has little control over the feed speed. At the same time, the router bit is trying to stay on the edge of the workpiece, thereby setting the situation where the cutting edges basically kick the workpiece along in the feed direction -- and out of the worker's control.