How the moon takes energy from Earth (if at all)? At least twice (on the comments of this answer, and mentioned in passing on this book), I have read that the moon takes energy from Earth due to tidal drag. The notion seems that: a) energy must conserve, b) if the Moon exerts tidal force on Earth, Earth must exerts reaction force on the Moon, and c) Earth loses energy due to tidal force, then, we can conclude that the Moon must gain the same amount of energy due to reaction force. This is what I suppose by taking quite a big reasoning leap, which I don't understand and doesn't sound plausible at all.
There is a similar argument made by this answer, which seems equally unappealing. I mean, angular momentum must conserve, but it doesn't necessarily have to go to the Moon.
I always had the impression that most of the spinning energy lost by Earth due to tidal forces (and the same happened to any tidally locked planet) is lost via viscous dissipation (heat, ultimately) on the fluids dislodged by the tidal force (atmosphere, oceans and, mostly, the liquid core (which also loses energy through the magnetic field it generates)).
The problem of conservation of angular momentum can be solved the same way as with any fluid vortex dissipated via turbulence: it is transferred to the smaller scales vortexes down to individual molecules, which on the average of the total rotation, they cancel out.
So, the question is, is there any meaningful amount of energy transferred from Earth to Moon due to tidal forces? How it affects Moon movement (or shape)? How this process works, exactly?
 A: Here is how I understand this.
Let me first discuss a case where the tidal effect is much stronger than in the case of the Earth tidal effect from the Moon.
Several of the Jupiter moon's that are close to Jupiter are subject to a very strong tidal effect. The moons are not in tidal lock. The tidal effect is so strong that the solid mass of those small moons isn't just subject to elastic deformation, there is plastic deformation. Jupiter is working those moons as if it is kneeding dough.
In the case of our Moon there is of course tidal effect on the water in the oceans. I'm not sure, but if I recall correctly the assessment is that the ocean tidal effects cancel out, because in many places the local tide is way out of phase with the overall tidal effect from the moon. 
(Later edit: in a comment contributor David Hammen has corrected me, stating that the vast majority of the tidal effect is in fact due to the ocean tides. I have edited this answer accordingly.)
The Earth tide response is not instantaneous. If the response would be instantaneous then the axis of elongation would continuously point straight to the Moon. However, there is a bit of a lag of the Earth tide, because of the sheer bulk. As seen from above the North pole: the Earth is rotating counter-clockwise. The axis of elongation, somewhat lagging behind the Earth rotation, is pointing somewhat to the left of the line to the Moon. 
When a celestial body isn't perfectly spherical the center of mass and the center of gravitational interaction are not exactly at the same point. 
To imagine why that is take the most extreme case: the point of gravitational interaction of a very long stick. The center of mass is the geometrical center, but since gravity falls off with the square of the distance the point of gravitational interaction is closer to whatever body it is interacting with.
The center of Moon gravitational pull on the elongated Earth is not perfectly at the Earth center of mass, but ever so slightly displaced. If the solar system would exist long enough eventually the Moon would slow down the Earth to tidal lock with the Moon.
The center of Earth gravitational pull on the Moon is not perfectly towards the Earth center of mass, but ever so slightly displaced. Over time this increases the velocity of the Moon, making it climb to a higher orbit.

Later edit:
About the magnitude of the effect: 
Kenneth R. Lang, Tufts University 
the Earth is currently slowing down at an average rate such that the days are getting longer at one second every 50,000 years.
A: Energy or momentum conservation is something I like to call "integral analysis" in the way that it compares the total of the initial state to the total of the final state, without explaining in detail how the transition happens. Hence - using energy and momentum conservation leads to exactly those explanations - earth has lost energy and momentum, and it has to go somewhere...
If you want to study how it happens, dynamics and forces are sometimes a better approach.
The simplest form I can think of: the moon excerts a braking torque on earth (through tides), and this torque needs a counter-acting, resulting on a torque of the same magnitude onto the orbit of the moon. Dividing this torque by the moon's (instantaneous) orbit radius, yields force on the moon, in the equatorial plane of earth.
A bit more elaborate how the forces are transferred: the tidal bulges are ahead of the moon, and then they pull the moon "further"... Here's a picture:
https://en.wikipedia.org/wiki/Tidal_locking
A: For the title question, the answer is actually rather simple. The same mechanism - the flexing of the planet due to the tidal deformation - that dissipates the Earth's rotational energy, also transfers some of it to the Moon. The way that happens is that, because it takes time for the Earth's solid material to deform under the gravitational stressing forces exerted upon it by the Moon, since the Earth is rotating in the same direction the Moon is orbiting but much faster (~86 ks rotation versus 2360 ks lunar orbital period), the induced "bulge" actually persists slightly and "gets ahead" of the Moon in its orbit. And the bulge exerts gravity of its own and hence actually tugs the Moon "forward", increasing its speed in its orbit and hence enlarging that orbit.
And while it may not seem like it should have much effect given the scales (Moon about 384 Mm distant from Earth, bulge height is only about 384 mm or $3.84 \times 10^{-7}\ \mathrm{Mm}$), this seemingly "negligible" force really adds up over the eons. While that's a factor of a good billion, we're talking times on the order of a billion years.
It adds up.
A: Yes, the slow transfer of angular momentum does increase the Moon's 
distance from Earth, moving it higher in our gravity well, and that means
it gains potential energy.   It also means that the tidal-locked lunar spin gets
slower, which is a loss of kinetic energy.

I always had the impression that most of the spinning energy lost by Earth due to tidal forces (and the same happened to any tidally locked planet) is lost via viscous dissipation...

The reason for torque in this situation is entirely in the time-delay of the
tidal bulges (part of which is solids flexing, part is ocean and even
atmospheric motions).  Without the energy loss mechanisms, those time delays
might not occur to the observed extent, it is possible to imagine
a liquid-drop pair of orbiting bodies that never 'lock'.
While energy can be turned to heat, however, angular momentum cannot;
the angular momentum of the earth-moon system is not subject to any
significant torques from outside that system (Earth is not likely to soon
tidal-lock to the Sun), so any spin angular momentum the Earth loses IS 
transferred to the moon's orbital angular momentum, less the attendant
change in the moon's spin angular momentum.
A: 
How the moon takes energy from Earth (if at all)?

Two key observed facts
The size of the Moon's orbit about the Earth is gradually increasing over time. This is an observed fact, a very precisely observed fact over the last 50 years thanks to retroreflectors left on the Moon by three American Apollo missions and two Soviet Luna missions.
The rate at which the Earth rotates is gradually decreasing over time. This too is an observed fact. This is a very precisely observed fact over the last 60 years thanks to atomic clocks, a less precisely observed fact over the last 400 years thanks to mechanical clocks, a historically observed fact thanks to recordings of lunar and solar eclipses by peoples in ancient Babylonia and ancient China, and a prehistorically observed fact thanks to tidal rhythmites that date back to many hundreds of millions of years ago.

Arguing from a point of view of energy doesn't quite make sense
It is impossible to have both the mechanical energy and the angular momentum of the Earth-Moon system be conserved in light of the slowing of the Earth's rotation rate and the increase in the separation between the Earth and the Moon. What is possible is to have one of the two be conserved while the other is not. It is of course also possible that neither mechanical energy nor angular momentum is conserved.
One reason that arguing from a point of view of conservation of energy doesn't quite make sense is that there is no such thing as a law of conservation of mechanical energy. Mechanical energy is only conserved in isolated non-dissipative systems. The putative mechanism behind the reduction in the Earth's rotation rate and the increase in the Moon's orbital distance are highly dissipative. Another reason that arguing from a point of view of conservation of energy doesn't quite make sense is that this would require the total angular momentum of the Earth-Moon system to increase as the Moon recedes from the Earth. That makes no sense at all.

Arguing from a point of view of angular momentum makes a good deal of sense
The total mechanical energy of the Earth-Moon system would necessarily decrease if the reduction in the Earth's rotation rate and the increase in the Moon's orbital distance result from transfer of angular momentum from the Earth's rotation to the Moon's orbit. In contrast to the increase in angular momentum that would necessarily result from conservation of mechanical energy, the decrease in mechanical energy that would necessarily result from conservation of angular momentum is not problematic. All that's needed is a mechanism that transforms some of the Earth-Moon systems mechanical energy into heat. That dissipative mechanism is observable at almost every shoreline.
What is needed then is a mechanism that explains the transfer of angular momentum from the Earth's rotation to the Moon's orbit. The standard explanation is the tides, and in particular, oceanic tides rather than the solid Earth tides.
The Earth tides can be ruled out as the dominant mechanism by the wide variations seen in tidal rhythmites in the rate of change in the Earth's rotation rate. The rate at which the Earth's rotation rate is currently decreasing is much higher than average. The Earth currently has two huge north-south barriers that interfere with a global flow of oceanic tides, the Americas and Afro-Eurasia. Moreover, the North Atlantic is shaped just right (or just wrong) so as to make for very large tidal dissipation losses. Models of the continents in the past indicate that even one large north-south barrier is uncommon. Two at the same time is very rare.

So what is the mechanism?
The mechanism almost certainly is an asymmetric time-varying gravitational effect that results from the tides. The standard explanation is that Earth's rotation pulls the tidal bulges that the Moon's gravity raises in the Earth's oceans slightly ahead of the Moon. It would be highly hypocritical of me to invoke that explanation due to my previous claim that those tidal bulges do not and cannot exist.
Note very well: I am not saying that the cause is something other than gravitational perturbations caused by interactions between the Moon and the Earth's oceans. Those interactions almost certainly are the ultimate cause. Proving that this is indeed the case will require highly accurate, time-dependent, global models of all of the key frequency responses to the tidal forcing functions, and then integrating the gravitational perturbations of these on the Moon over a sufficiently long period of time. The science is not there yet -- but it will be.
