Why does light travel at the same speed when measured by a moving observer? I know the hypothesis that the light speed is constant is retained by experiments. But is there any theory explaining why the light speed is constant no matter how an observer moves relative to light?
My question is, specifically: Suppose an observer $O$ launches a light and $O$ starts to move at the same time with a uniform velocity $v$ in the same direction that light points. Then why $c$ is still the light speed that $O$ will measure rather than $c-v$?   
 A: The answer is - velocities just do not add like that when we are getting close or even equal to the speed of light $c$. The whole body of Special theory of relativity concerns itself with what are the consequences of postulating the speed of light to be constant in all frames of reference. 
That is, we find transformations between frames of reference by postulating the speed of light being constant. On the other hand, once we have constructed the theory, the fact that secures that we see light in vacuum to move constant in every frame of reference is the Lorentz transform. The Lorentz transform tells us how observers at different relative speeds observe the same things differently.
To demonstrate this on your example, we have a ray at speed c in the $x$-direction so that we see
$$\frac{d x_{ray}}{dt} = c$$
Then we "boost" ourselves by a velocity $v$ also in the $x$-direction. Our time and $x$-direction are suddenly transformed so that we observe
$$x \to \frac{x-vt}{\sqrt{1-v^2/c^2}}, \; t \to \frac{t-vx/c^2}{\sqrt{1-v^2/c^2}}$$
which applies also to the observed $x_{ray}$. We have to differentiate to obtain the velocity of the ray
$$\frac{d x_{ray}}{dt} \to \frac{dx_{ray} - v dt}{dt - v dx_{ray}/c^2} = \frac{dx_{ray}/dt - v}{1 - v/c^2 (dx_{ray}/dt)} = \frac{c - v}{1-v/c} =c$$
So, the speed of light $c$ is exactly conserved by the transform. If you don't understand the math, the intuition is just that time and space dilate and contract exactly in such a way that the speed of light is observed to be the same from any point of view.

EDIT :
If we want to know why c is always invariant, we are actually asking why do physical observers transform according to the Lorentz transformation? Or, why does space and time dilate/contract etc. etc. We might find a deeper structure which explains the Lorentz transform, but it will once again be based on some suppositions. And then we can ask about these suppositions. It then all boils down to a "Why these laws?" questions. Science just does not provide complete answers to such questions.
As to deeper structure, this paper by Mitchell Feigenbaum is so far the only one I know which shows (not entirely clearly, but it is there) that under the purely Galilean assumptions of isotropy and "homogeneity" (leading to linearity) conserved by transforms, we necessarily get only one possible family of transforms with a single parameter $1/c^2$. Setting $1/c^2 = 0$ we get the Galilean transform, setting $c$ to be the speed of light in vacuum, we gain the Lorentz transform. I.e., the fact that we do have a transform conserving isotropy and homogeneity under any observations causes that there is a transform giving rise to a universal speed limit $c$ (which might be even $c = +\infty$). 
On the other hand, once we allow for the four-dimensional Newton's law, we get that for a particle with rest mass $m_0 \to 0$ and non-zero kinetic energy, this particle will always attain a velocity $\to c$. That is, if we know a particle with zero (or almost zero) rest mass, we identify this speed limit by measuring it's velocity. So this is "why" the universal speed limit is the speed of the assumed massless particle - the photon. 
We could argue that since Nature does not allow infinities to arise, the universal speed limit is a consequence of having a massless particle. But this is a rather weak and cyclical argument. I think I can't give you any more definitive answer.
A: A personal point of view is that you may consider that Lorentz transformations apply primarily on  momenta, and not primarily on (infinitesimal or not) space-time coordinates.
This is, of course, a "strong" postulate.
If you assume (some additional postulates are needed there) that transformations are linear, and that there is a rotation invariance, you are going to study "boost" transformations : $\begin{pmatrix} p'_z\\E'\end{pmatrix} = A(v) \begin{pmatrix} p_z\\E\end{pmatrix}$. You may show that, because  $A(v)A(-v)=1$, $det A(v)=1$. Supposing a group structure, you finally are looking at the one-dimensional subgroups of $SL(2, \mathbb R)$, which are : 
$$\begin{pmatrix} \lambda&\\&\lambda^{-1}\end{pmatrix}\quad \begin{pmatrix} 1&v\\&1\end{pmatrix}\quad \begin{pmatrix} 1&\\v&1\end{pmatrix}\quad \begin{pmatrix} \cos \theta&-\sin \theta\\\sin \theta&\cos \theta\end{pmatrix}\quad \begin{pmatrix} \cosh \theta&\sinh \theta\\\sinh \theta&\cosh \theta\end{pmatrix}$$
If you add additional postulates that, in a boost transformation, energy and momentum must change, that there exist a transformation which puts the momentum  to zero ,  and  that, if the energy is positive for an observer, energy will be positive for all observers, the first $4$ one-dimensional subgroups of $SL(2, \mathbb R)$ are excluded, and the last dimensional subgoup corresponds to a Lorentz transformation.
A: "The speed of light is constant, no matter how an observer moves relative to the light." 
This is true. The light is moving independently, thus an observers actions of movement, make no changes to it.
However, the interesting part is that the speed of light is always "Measured" as the speed of light no matter how an observer is moving relative to that light. This includes an observers movement in any particular direction, meaning any direction relative to the direction of the light itself.
The key to understanding this is to view reality in a 4 dimensional manner.
All mass particles are constantly in motion at the speed of light within that 4 dimensional environment known as Space-Time. All that can be done is change the direction of that constant motion. Thus if one is at true rest in space, the entire constant motion is now across the dimension of time only.
To perform a measurement, one must have a complete measurement instrument.  This will include one instrument that measures spatial length (Ex. Ruler ). Two synchronized clocks are also required. The clocks are placed at the opposite ends of the ruler. 
If light, from any source, passed from one end of the ruler to the other in either direction, while the ruler is at true rest in space, a measurement of the speed of that light will indicate that the light traveled at the speed of 300,000 kps.
If the measurement instrument was now in motion across space, then a change of direction of constant motion across Space-Time has occurred. 
In turn, there is a reduction of motion across the dimension of time, thus both clocks are now ticking slower. Rotation has also taken place, thus the ruler begins to extend across the dimension of Time and thus less of it extends across Space, hence we encounter spatial ruler length contraction.
Also, due to this extension across time, the clocks at the ends of the ruler are no longer synchronized, since they are now positioned at different points in time.
If once again, light from any source passed from one end of the ruler to the other in either direction, a measurement of the speed of that light will still indicate that the light traveled at the speed of 300,000 kps. This is due to the multiple changes of the measurement instrument itself. 
Thus in both cases we measure the speed of light, as still being the speed of light. 
Thus in turn we can not determine if we are at true rest in space or not, since the results are the same whether we are at true rest, or not. Thus we also can not determine what direction we are traveling in within Space-Time.
On top of that, if we had 2 identical spaceships, with one being at true rest in space and the other being in motion across space, to those onboard the one at rest, it seems as though the other spaceship has shrunk in spatial length, and that their onboard clocks are ticking slower.
However, due to the change of the measurement instruments onboard the moving spaceship, it appears to them as though the space ship that is at rest in space has shrunk in spatial length, and that its clocks are ticking slower. Thus from their point of view, it appears to be the spaceship at rest that is actually in motion, rather than their's.
What is important is to understand why in both cases it always seems to be the other guy who has shrunk in length, and the other guy whose clocks are ticking slower.
This rule applies to two bodies, be it one at rest in space and one not, or two bodies that are both moving across space but moving a different velocities.
Thus the absolute point of view can not be detected, thus absolute was dropped from being seen as important, thus, in most cases these days, violent opposition continues against the absolute even though such an intuitive 4 dimensional point of view explains it all and produces the exact same equations.
