What is the deep reason of length contractions and time dilations? In the theory of relativity, a spacetime can have length contractions or length expansions and time can have time dilations or expansions.
In theory of special relativity, any physical object with constant velocity or any inertial coorinate system, can have length contraction and time dilation such that:
$$ (t'/ t_0) = \sqrt[]{1-(v/c)^2} $$
$$ (s'/s_0) = 1/\sqrt[]{1-(v/c)^2}$$
This property is also symmetric between to observers that have relative constant velocity - both observers observe the same phenomena in the other.
and in Genral relativity, for example in the Schwarzschild metric object in the gravitation field has time dilation:
$$ (t'/t_0) = \sqrt[]{1- (2GM/rc^2)}$$
compared to a clock that is far away from the gravitation field.
Usually i have read that these properties belong to the geometry of spacetime. The geometry of the empty space for example follows so called Lorenzt transformations between two origin-centered inertial reference frames, other having relative velocity v other having v= 0 is:
$$ t' = u(t - \frac{vx}{c^2})$$
$$ x' = u(x - vt)$$
$$ y' = y $$
$$ z' = z $$
whwere u is Lorentz gamma factor $$ u = \frac{1}{\sqrt{1-(v/c)^2}} $$
Also the observed local velocity of ligth is always constant for all local observers:
$$ c'= c $$ 
However, In the gravitation field, different observers do not agree that the velocity of light is constant globally, in large path intervals.
These equations that describe the geometry of spacetime does not yet tell what is the reason why spacetime has this kind of geometry.
1 Is there any theory or insight what is the reason behind these time dilations and length contractions? 
This has been answered: Lorentz covariance; Lorentz covariance is a property of spacetime that it obeys the Lorentz group transformation equations.
And is there any idea what makes the velocity of light is always constant in empty space?
(these following related questions 2 and 3 are asked to me to better to move under new question:) (edited 22.12.2015) 
2 can Lorentz covariance be a result of some kind of dynamical or nondynamical but active process or mechanism that is going on in the spacetime or in space in small length scales and short time scales?** (edited 22.12.2015)
3 Is there another deep explanation what is behind "Lorentz covariance"?** (edited 22.12)
 A: 
What is the deep reason of length contractions and time dilations?

The wave nature of matter.  

In theory of special relativity, any physical object with constant velocity or any inertial coordinate system, can have length contraction and time dilation such that $ (t'/ t_0) = \sqrt[]{1-(v/c)^2} $ [and] $ (s'/s_0) = 1/\sqrt[]{1-(v/c)^2}$. This property is also symmetric between two observers that have relative constant velocity - both observers observe the same phenomena in the other.

Yes, the Lorentz factor is derived from Pythagoras's theorem and it "works" because of the wave nature of matter. See the simple inference of time dilation due to relative velocity on Wikipedia. The hypotenuse of a right-angled triangle represents the light path where c=1 in natural units. The base represents your speed as a fraction of c. The height gives the Lorentz factor, then we use a reciprocal to distinguish time dilation from length contraction. 

Public domain image by Mdd4696, see Wikipedia 
This applies to electrons and other particles because of the wave nature of matter. We can create electrons (and positrons) out of light in pair production, we can diffract electrons, and then we can annihilate the electrons with the positrons to get light back. Think of the electron as light going round in a circle, then turn the circle on its side like so | and then think in terms of a helical spring /\/\/\/\/\.   

in General relativity, for example in the Schwarzschild metric object in the gravitation field has time dilation: $ (t'/t_0) = \sqrt[]{1- (2GM/rc^2)}$ compared to a clock that is far away from the gravitation field.

No problem. Think in terms of an optical clock. But note that two observers at different elevations both agree that the lower observer's clock is going slower. The time dilation isn't symmetrical like it is for SR observers. 

Usually I have read that these properties belong to the geometry of spacetime...

IMHO you'll hear that for GR time dilation, but not for SR time dilation. 

Also the observed local velocity of light is always constant for all local observers...

Of course it is. Because of the wave nature of matter. Have a read of The Other Meaning of Special Relativity by Robert Close. It explains why the SR observer always measures the local speed of light to be the same. 

However, In the gravitation field, different observers do not agree that the velocity of light is constant globally, in large path intervals. These equations that describe the geometry of spacetime does not yet tell what is the reason why spacetime has this kind of geometry. Is there any theory or insight what is the reason behind these time dilations and length contractions?*

Yes. General Relativity. But you have to read the original material. A lot of modern authors appeal to Einstein's authority whilst flatly contradicting him, then they'll start telling you about the parallel antiverse and quantum consciousness and other such nonsense. This is what Einstein said, see the second paragraph, which means you have to be cautious about c=1:  


Can it be a result of some kind of dynamical or nondynamical but active process or mechanism that is going on in the spacetime? 

Yes, but in the space rather than the spacetime. Maybe you need to ask a separate question about that.

And is there any idea what makes the velocity of light is always constant in empty space?

Yes. The (coordinate) speed of light depends on the "metrical qualities" of space. If these are uniform there's nothing to affect the speed of light. A gravitational field is a place where space is "neither homogeneous nor isotropic". 
