energy-conservationforcesfree-body-diagramnewtonian-mechanicswork
If you lift a book from the floor to a shelf, $K_f-K_i=0$,because the final and initial speed are 0.
So from the work–kinetic energy theorem the work done by external forces should be $0$, so there is no energy transferring mechanism here, since work done by external forces is $0$, and there is no other energy transferring mechanism here, so how does this the system gain energy in form of potential energy?
The total work can be split up into two parts:
$$W_{net} = W_{conservative}+W_{non-conservative}.$$
With the conservative part you can associate a potential energy:
$$W_{conservative}=-\Delta PE$$
(this is in fact the definition of a conservative force) so that the Work-Energy theorem becomes
$$W_{non-conservative}=\Delta KE + \Delta PE = \Delta E.$$
This is another way of writing the Work-Energy theorem and in my mind it's a little bit clearer. Restated, the work done by non-conservative forces is equal to the overall change in energy of the system.
For example, work done by friction is negative, so it dissipates energy away from a system.
On the other hand, gravity is a conservative force. Imagine the motion of a falling ball. Unless something doing work on the ball to slow it down (for example, air) the ball will speed up as it falls. In this case, the equation
$$W_{gravity} = -\Delta PE = \Delta KE$$
is equivalent to that statement. (As the potential energy becomes more negative, kinetic energy becomes more positive.)
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
The system which gains the gravitational potential energy is the book and the Earth not the book alone.
You do work increasing the separation between the book and the Earth and the result is that the book and the Earth have more gravitational potential energy.
The book alone as the system has two forces acting on it.
So the net force on the book is zero, the net work done on the book is zero and so the change in kinetic energy of the book is zero.
Put another way the positive work done by you in lifting the book is equal to the negative work done by the gravitational attractive force.