I never particularly liked the textbook description of this topic.
The key concept is that there's a quantity called "energy" that we've decided is useful and can't be created or destroyed$^1$, it can just change "types". This is useful because if you choose a system you want to study carefully, you can learn a lot about its behavior from energetic considerations.
Broadly speaking, there are two types of forces, conservative and non-conservative forces. A conservative force is one for which a potential can be defined, and with that potential comes an associated potential energy. For instance, for gravity I can define a potential:
$V(\vec{r}) = -\frac{GM}{||\vec{r}||}$
And there's an associated potential energy:
$U(\vec{r}) = -\frac{GMm}{r} + U_0$
So gravity is a conservative force. The abstraction is that by lifting an object in the gravitational field, I do work and store energy in the field. The field can later release the energy, and no energy is "lost".
Friction is really complicated. It can be, and is, modelled simply, but the process at a microscopic level involves rapidly created and destroyed chemical and/or physical bonds and is not fully understood. When work is done involving friction, we decide not to describe this as energy being stored in some sort of "friction field" (we would need to define a friction potential, and there's no obvious way to do this). Instead, we describe the process by saying that the energy is dissipated by friction, lost as vibration or heat (note that these are both a type of kinetic energy - heat is just a description of the average motions of a collection of particles). The important difference as compared to a conservative force is that heat, for instance, cannot be released from a surface to make a block slide faster. The energy dissipated into heat is "lost" from the system of a block sliding on a rough surface.
With all that in mind, tackling energy conservation problems just takes a bit of practice. My advice would be to forget about the formulae a little bit. Instead, look at the system you're considering and try and account for all the relevant forms of energy, and all possible exchanges/transformations between types of energy. The big "trick" is to define the extent of your system carefully. Students I've taught seem eager to add a thermal energy term to their analysis of problems involving friction, but this is often not a useful exercise. If it's sufficient to know that some energy was dissipated away as heat, you can just include a term in the math that expresses that the system lost, for example, $\mu_kN\Delta x$ of energy.
If I had to break it down into steps, I'd say:
1) Pick the initial state of your system, tally up any potential and kinetic energy.
2) Pick the final state of your system, tally up any potential and kinetic energy.
3) Go through the processes that occur between the initial and final state. Do any of them dissipate energy from the system? Or inject energy?
4) Add everything up (being careful about the sign of each term). Any difference between the initial and final energy of the system should be accounted for by energy injected into or dissipated out of the system in between.
$^1$ At least in simple physics... you can formulate theories/models where energy is created/destroyed, but this is only done if there is some advantage to doing so.
I agree with CuriousOne that the example is more confusing than helpful, but this is the way I would explain it.
Suppose you take a spring, place it on the ground then compress it. If you now suddenly let go of the spring it will rebound and bounce upwards off the ground:
The spring clearly has work done on it because its kinetic energy increases and that increase must have come from somewhere. However the ground can't have done any work on the spring because the ground hasn't moved. It should be obvious that the potential energy in the compressed spring has been converted into kinetic energy of the uncompressed spring - in effect the spring has done work on itself. This is what your book means by:
transfers of energy from one type to another inside the object
i.e. potential energy of the compressed spring has been converted into kinetic energy of the uncompressed spring.
In the case of the skater the skater's arms correspond to the spring and the rail corresponds to the ground. The skater's arm isn't a spring, of course, because it's chemical energy not potential energy being converted to kinetic energy by the skater's muscles. Nevertheless the same principle applies.
Best Answer
A force may be thought of as any influence which tends to change the motion of an object. Note that I have emphasized the term "tends", because a force does not necessarily result in the acceleration of an object. Only a net force causes acceleration.
For example, I can exert a force on a wall. But if the ground exerts an equal and opposite force on the wall, the net force on the wall is zero and it will not accelerate.
By Newton's third law the wall exerts an equal an opposite force on me. But if the static friction force between my feet and the ground is equal and opposite to the force the wall exerts on me, the net force on me is zero and I do not accelerate. .
Net work is done on an object that is subjected to a net force causing the displacement of the object transferring energy to the object. Again the key is the object must be subjected to a net force.
Taking the above example of me and the wall, the net force applied to each of of us was zero. There is no displacement of either, and no work is done on either. Again the key is an object must be subjected to a net force in order for work to be done on or by the object.
But just because an object is moving does not mean it is subject to a net force and that work is being done, as discussed below.
If the object is moving due to a constant net force applied in the direction of the movement, then the work done on the object in moving it a distance $d$ is $W=Fd$. And what that work does is to increase the object’s kinetic energy by an amount equal to the net work done on the object.
But you need to be careful. An object can be moving without a net force. This is the case when an object moves at constant velocity.
The work energy theorem states that the net work done on an object equals its change in kinetic energy. If the velocity of an object is constant, there is no change in kinetic energy and the net work done on the object is zero.
Hope this helps.