When landing, the plane is steered by the rudder. As the plane slows, the rudder become ineffective, and the pilots steer the plane like a tricycle with the nose wheel. When conditions are extremely slippery, the nose wheel may not have enough adhesion to steer the plane. When exiting the runway, unless the plane is moving at a very low speed - perhaps no faster than a person walks - the plane may skid off the slippery pavement onto the unpaved area alongside the taxiway.
Anxious fliers need to recognize that, since this happens at slow speed, there is no threat to safety. The worst result is embarrassment to the pilot, and delay in getting the passengers to the terminal. The plane has to be moved by a tug. In some cases, the passengers stay aboard as the plane is towed; in other cases, they deplane and are bussed to the terminal.
In slippery conditions, pilots generally rely on after-landing reports by other pilots. When pilots report conditions are too slippery to continue operating, the runway is shut down so sand can be applied to provide traction.