In car driving, why does wheel slipping cause loss of control?
Because friction is your method of steering! (- and of braking and accelerating.) As @MasonWheeler comments:
This is such an important principle that there's a special name for it: in the specific context of using applied friction to direct motion, friction is also known as traction.
Turning / steering
Friction is what makes you turn left at a corner: you turn the wheels which directs the friction the correct way. In fact, by turning your wheels you turn the direction of friction so that it has a sideways component. Friction then pushes your wheels gradually sideways and this results in the whole car turning.
Without friction you are unable to do this steering. No matter how you turn your wheels, no force will appear to push you sideways and cause a turn. Without friction the car is drifting randomly according to how the surface tilts, regardless of what you do and how the wheels are turned.
Braking and accelerating
Accelerating and braking (negative acceleration) requires something to push forward from or something to hold on to. That something is the road. And friction is the push and the pull. No friction means no pull or push, and braking and accelerating becomes impossible.
So, friction is very, very important in any kind of controlled motion of vehicles that are in touch with the ground. Even when ice skating, you'd have no chance if the ice was 100% smooth.
It should now be easy to grasp that it's a problem to go from static friction (no slipping of the tires) to kinetic friction (the tires slip and skid), simply because kinetic friction is lower than maximum static friction.
If you brake e.g., it is better to have static friction, because it can reach higher values than kinetic friction and thus it can stop you more effectively.
While the handling difference has to do with differences between static and kinetic friction, it is not the difference between the static and kinetic coefficients of friction that explains loss of control, but the direction of the friction force.
As others have explained, in rolling without sliding the contact point between the road and the tire is stationary. In stationary friction, the friction force is directed in whatever direction is needed to prevent sliding from happening. When you turn the wheels of your car, movement forward happens in the rolling direction without the contact point sliding, and the static friction force is mostly directed perpendicular to the rolling direction, providing the centripetal force needed to make your car go around the corner.
But the moment the tires start slipping on the road, the contact point is no longer stationary. And when there is sliding at the contact point, the kinetic friction force is directed opposite the direction of sliding. Even if the total friction force stays the same, all of a sudden it is no longer providing the centripetal acceleration needed to go around the corner, but instead linearly decelerating your car. Which means you run off the road at a slower speed than you were cornering, but you still end up in a ditch.
A similar situation can happen when braking in a strong cross wind. While you are rolling without sliding, the stationary friction counters the sideways force of the wind to avoid sliding. But if you lock your wheels and start sliding, the kinetic friction force is applied against the direction of sliding, that is, mostly backwards, and without friction to oppose it the wind all of a sudden can push your car into another lane.
Or when trying to pull a cork from a bottle: if it is too hard to pull it straight out, you may find it easier to make it rotate. The moment it starts rotating, even the slightest force applied upwards will make the cork move out, because most of the friction is happening in the perpendicular direction.
On ice, as the surface is so slippery, you may find that the tyre is always slipping to some extent. Because the tyre now rotates faster than needed for the speed of the car, a point on its circumference will cover a greater distance than the car. In other words, instead of being stationary with respect to the ground, the contact point moves.
Now how does this affect control of the car?
If you look at graphs of friction vs applied force, you'll see that friction increases with force, until movement occurs. Then it suddenly drops.
To explain this, you need to realise that, as you apply more force to an object and there is no movement of the object, the friction force has to increase with the applied force: if the two were not equal the object would start to move, in accordance with Newton's laws of movement. Once the force gets past a certain point, friction is no longer able to resist it and the object starts to move. At that point the friction force drops dramatically.
You can understand why when you think of all the small imperfections of the 2 surfaces (the object and the substrate). These imperfections tend to mesh into each other, preventing the object from moving. Once the object is moving, the bumps no longer have time to fall into the depressions; instead they just skid over the top. Hence friction decreases.
What does all this have to do with a tyre? You need to realise that the point of contact between the tyre and the road is stationary with respect to the road. The contact point is different from moment to moment, but every such point is stationary on the road. It's only when the wheel locks up or slips that the contact point actually starts to move. At that point you loose traction.
Thus, on ice, because the contact point with the ice is not stationary, sliding friction decreases and you lose control of the car.