Why is ice a problem for airplanes?
Ice reshapes the surface of the lift-producing parts of the airplane: the wings and the tail. That roughness is enough to change the aerodynamics of the wing such that there's more drag and less lift.
The amount of lift a wing creates depends on the relative angle that the airstream has to the airfoil. As you increase that angle—the angle of attack—you generate more and more lift. But at some point air cannot flow over upper surface, and you have aerodynamic stall. The point at which aerodynamic stall takes place has to do with the contour of the airfoil. If the surface is contaminated with slight roughness—sandpaper roughness—it will reduce the lift and change the point at which stall takes place.
For scheduled air carriers [including commercial passenger airlines] icing has been a contributing factor in 9.5 percent of fatal air carrier accidents.
How are pilots trained to handle aerodynamic stall?
As you go through pilot training—without icing involved—you practice wing stalls. You bring the nose up and the airplane shakes around because of separated flow. To recover from that, you push the nose down to reduce the angle of attack on the wing and recover. What happens with ice is same principle, but it is happening at a lower angle of attack.
Why would having a plane on autopilot interfere with a pilot's ability to prevent stall?
If a person is hand-flying an airplane and the airplane has a reversible control system, then for every action the pilot makes on the control there is some reaction on the control. There is the ability for the airplane to talk to you.
When the autopilot is engaged, that information isn't being passed onto the pilot. The National Transportation Safety Board recommends against flying with autopilot under icing conditions. [The Federal Aviation Administration recommends against autopilot only under severe icing conditions.] Companies make their own choices on how to present that information to the pilot.