Why do wheels spin? In my physics textbook it says cars move because the wheels exert a backward force on the ground and the ground exerts an equal and opposite forward force on the car. If the wheel spins clockwise, then there is a resultant clockwise moment on the wheel due to the force from the engine. My question is, why doesn't the equal and opposite force from the ground create an anticlockwise moment on the wheel which cancels out the clockwise moment and hence stop the wheel from spinning, instead of making the car move forward?
 A: The engine provides a force to turn the wheel ( in physics lingo torque ) this causes the wheel to turn counter clockwise. Now look at the point of the tire that touches the ground. This point on the ground experiences one force as the wheel tries to turn - static friction (not kinetic, but that's an entire discussion in itself). This force applies a push that is proportional to the weight of the car (assuming the car is on a flat surface). This pull on the wheel creates traction (a difference in forces) which allows the wheel to turn and the car to move. Using Newton's third law which you cited isn't the correct line of thought as it means that the force friction exerts on the wheel will be equal to the force that the earth experiences from the wheel. I hope my explanation helps clarify this subject.
A: 
...why doesn't the equal and opposite force from the ground create an anticlockwise moment on the wheel which cancels out the clockwise moment and hence stop the wheel from spinning, instead of making the car move forward?

It does!  That is, the moment on the wheel from the road does (almost) stop the angular acceleration of the wheel.  The approach you describe is standard practice in understanding components of more complicated systems, as I explain below, so this is a good example to work that out.
I'm sure you've seen a car wheel without the friction of the road, for example on a lift, on ice, or when a car is stuck in the mud.  There you can see the angular speed of the wheel ramps up extremely fast.  That's what a wheel looks like without the torque from the road.  But on the road it doesn't do this.  That is, the force from the road balances almost the entire torque on the wheel bringing it close to zero.  There is a little bit of torque that doesn't balance which is why the wheel spins (and has angular acceleration) when the car is moving, but, by far, most of the torque is cancelled out by the mechanism you suggest.
edit:
A more detailed understanding is easy to see if one writes down the equation of motion for the car and wheel.  Here 


*

*$M$ is the mass of the car

*$m$ is the mass of the wheel

*$R$ is the radius of the wheel

*$I$ is the moment of inertia of the wheel, which I'll assume to be a disk, so $I = {1\over2}mR^2$

*$a$ is the acceleration of the car

*$\alpha$ is the angular acceleration of the wheel, so $\alpha = a/R$

*$\tau_A$ is the torque supplied by the engine at the axle


Keep in mind that the mass of the wheel is much less than the mass of the car, something like $M = 100m$. Then, the torque at the axle needs to both accelerate the car and make the wheel spin:
$$\tau_A = I\alpha + MaR$$ or $$\tau_A = ({m\over2} +M)Ra$$
So one can look at the above equation in several ways.  It's true to say that part of the axle torque $\tau_A$ goes to spinning the wheel and part to pushing the car forward.  But it's equally correct to describe a net torque on the wheel, $\tau_{NET}$, that's the axle torque less the torque caused by the reaction force, $MaR$, or, $$\tau_{NET} = \tau_A - MaR = {m\over2}Ra$$  That is, if the wheel is going to move in a certain way (here have acceleration $a$), then the net torque on the wheel must be consistent with that motion, so it's good to see here that it is.  But it requires the understanding that you described where the reaction force counters the drive torque.
Since $m\ll{M}$, we have $\tau_{NET}\ll{\tau_A}$.  That is, the reaction force almost completely cancels out the input torque, leaving just a small net torque on the wheel.  That's expected since the wheel is much lighter than the car.  The result of this is that the wheel moves appropriately to roll with the acceleration of the car, but it spins much much slower than it would if all of the torque $\tau_A$ went purely to making the wheel spin (eg, on ice, etc).  So you're right that the reaction force creates a counter torque which cancels the drive torque and stops the wheel from spinning... it's not a complete cancellation, but almost complete, i.e, to within about $m/M$.
By the way, this chain of reasoning is actually fairly common.  For example, consider a lever on a fulcrum with a mass on one end.  When the lever lifts the mass you could ask, "doesn't the mass provide a reaction force that makes the lever not move?"; and you could analyze the whole system (lever + mass), or you could just look at the lever and find that the mass does provide a reaction force but when you work it out fully, there's just enough net force on the lever to allow it to move.
In general, this approach is important when comparing components as parts of a system vs in isolation.  Specifically, forces that were action forces (where, say, part A pushes into part B) become reaction forces when the part is isolated (where now the force on A from B are considered to hold it in place -- as B would have but now it's been removed), etc.  (Also, Newton's third law is the key to understanding these isolated situations, and is standard practice and understanding, which is why I don't like the other answer which says that the 3rd law is not the correct line of thinking.  It is exactly the correct line of thinking, and one just needs to work it through consistently.)
A: This is my understanding:
The car and the Earth apply equal and opposite forces on eachother by the torque of the engine trying to turn the wheel. This causes the Earth to move backwards a little bit, and more significantly, causes the whole car to move forwards which in turn makes the wheels move.
I think what makes this case with cars confusing is that it is the wheels themselves that apply the force in order to move the vehicle forward. If you think of a car that accelerated with the help a giant fan on the back, it is easier to see that the wheels turn simply because the car is being accelerated into motion by the fan.
In summary: The torque from the wheels creates a force between the Earth and the whole car which makes the car move and it is this movement that turns the wheels.
A: Imagine that the car is moving to the right and the wheels are rotating clockwise.
The no slip condition is $v=r \omega$ where $v$ is the linear speed of wheel axle to the right, $r$ is the radius of the wheel and $\omega$ is the clockwise angular speed of the wheel.
Now suppose that the wheels are made to rotate faster (spin) which means that the angular speed of the wheels is too large or the linear speed of the axle is too small for the no slip condition.
So to get to the no slip condition again the angular speed of the wheels must decrease and/or the linear speed of the axle must increase.
A forward frictional force on the wheel will exert an anticlockwise torque on the wheels thus trying to reducing the angular speed of the wheels whilst at the same time exerting a forward force on the axle thus trying to increase the linear speed of the axle.  
This continues until the new rolling without slipping condition is achieved. 
A: TLDR: In the analogy of a car moving on a road, it is not exactly the torque that is making the wheel spin, at least not directly (because the torque actually does get cancelled out by the friction, like you said). The torque first creates momentum, and it is actually the momentum that causes a reaction force on the wheel, which is what makes it spin.
This is a rather interesting topic which is actually more complex than is immediately apparent. I will try to elaborate on it as best as I can.
Firstly I will start by stating these two things:

*

*In Newton's Third Law the forces are equal and opposite, but they don't cancel each other out because the action and reaction force never act on the same object. This is a common confusion. If there is an action force on an object, the reaction force will always act on the object that is applying the force. If forces are being cancelled out, the only way that can happen is that if there are multiple pairs of action and reaction forces acting in the system.


*In this example there are three distinguishable systems in play here, first is the wheel and the engine/car, second is the wheel and the ground, third is the car as a whole and the ground. You have to consider all of them to get the full picture.
Consider a a car moving forward in a road (left to right from your point of view). The engine is first applying a torque on the wheel which is creating a tendency to spin clockwise, but since it is in contact with the ground, the spin is constrained and the wheel will apply a backward frictional force on the ground, which in turn will apply an equal and opposite reaction force on the wheel, creating an anticlockwise torque and this will in fact cancel out the spin initially (interestingly, that is also why the point at which the wheel is in contact with the ground always has a velocity of 0 relative to the ground). But then this will cause the forward frictional reaction force on the wheel by the ground to translate to the entire body of the car (because the wheel in this stage is basically acting like an object that is fixed to the car because there are constraining forces that are in equilibrium, without which the wheel would spin freely but the reaction force would not translate to the car). This then causes the car to have a forward momentum, and since the car and wheels are in contact with the ground, this momentum will again apply a frictional force on the ground, but this time in the forward direction, and then the reaction force from the ground will act on the wheel in a backward direction, giving it a clockwise spin when the car is moving.
