Why Speed of light in vacuum remains constant? I know that the speed of light remains constant irrespective of whether the light source is moving or not. But it's very difficult for my brain to visualize it. Light has dual characteristics, i.e. of both waves and particles, thus how can it be possible if a light source is moving near the speed of light that its emitted light will not reach more than $c$? (I know, as per Einstein, time slows down for moving objects.)
 A: You can see the mathematics about the velocity addition formula and say, wow this is how it works. But you are asking why?
We use mathematics to describe the real world, and not the other way around. And you are correctly asking for a down-to-earth explanation why reality is like that. Your intuition tells you that if the light source moves in space at speed let's say 0.9 c, then the emitted light from the source should move at a speed that adds the speed of the lightsource (that sped up to 0.9 c) and the speed of light, because the light already has a headstart.

In relativistic physics, a velocity-addition formula is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different frames. Accompanying velocity addition is a kinematic effect known as Thomas precession, whereby successive non-collinear Lorentz boosts become equivalent to the composition of a rotation of the coordinate system and a boost.

https://en.wikipedia.org/wiki/Velocity-addition_formula
You are thinking about this the wrong way. First, after the big band, there were only massless particles (sea of photons), all moving at speed c in vacuum. This was the only speed.
Now to slow down from this speed in the spatial dimensions, you need to gain rest mass. Some particles (and objects) gained rest mass and their spatial velocity decreased to slower than c. 
Now in your case one of these objects moves at let's say 0.9 c, so relative to the only speed for massless particles, it is 0.9 times slower.
This object emits light. These massless particles as they are emitted, start moving at the only allowed speed they can, c.
In this case, there is no wrong intuition. The massless particles that are emitted cannot go faster than c, since this is the only speed. Everything else (that has rest mass) moves relative (slower) to them.
A: Please note that I have limited scientific knowledge. This is only a popularized explanation that helped me understand that concept. Take this for what it is, it may be inaccurate or very simplified.
The problem is that our real life definition of "speed" is flawed when talking about photons.
Space-time has 4 dimensions, time being one of them. Every single object (not only photons !) in this space-time moves at the same speed, c. What this means if that the length of the vector used to represent this speed is always c. Its direction varies, though. So the length of its components in each dimension vary. Let's call these x, y, z and t. What is really important here is the value of t.
For all real life objects, t values are extremly similar and extremly close to c. They "move" at almost full speed in time, but very slowly in space. That's why it makes sense to consider "time speed" as a constant and measure how x, y and z vary depending on the position in time. That's what our real life definition of speed is. That's why we get the illusion that objects can move at different speeds. They do, but in 3D space, not in 4D space-time.
This definition of speed does not make any sense for photons, because in their case t = 0 ! Our definition considers the position of photons in the 3 dimensions of space depending of their position in time. But their position in time never changes ! The idea of "speed of light" based on what we usually mean by "speed" is absurd, that's why it seems incoherent.
Here is my source, it's a video in french but has english subtitles : https://www.youtube.com/watch?v=kELX0GEQ0H0
A: Imagine people flying over a lake, tapping the water quick enough so only a ripple emanates from it. It doesn't matter how fast or slow people are going, the speed of the ripple stays the same. You are in the lake's frame, other people would see it from their own lake's frame, everyone has their own lake that they see others touch (this may seem weird but relativity describes reality for us, not the other way around). And if you want a particle representation, you can think of dividing the ripple into wave packets (again, this may seem weird but quantum mechanics gives us a new picture of reality, what we see is described by it, not the other way around).
Maxwell's equations predict a speed of light from a source, and Einstein showed that the laws of physics are the same for any inertial frame, thus the speed is the same for everyone, no matter how fast or slow they are moving. 
A: 
Why is the speed of light independent of the speed of the source?

There is nothing peculiar at all here. This is not where special relativity or quantum mechanics comes in. This is, in fact, the standard behavior of all wave phenomena, i.e., the speed of a wave depends only on the medium and not on the source. For example, the speed of the sound wave from a loudspeaker traveling at $50 \text{ m/s}$ 
and that of the sound wave from a loudspeaker at rest are both the same. This is just plain old wave mechanics, nothing peculiar here. 
This is very intuitive to understand too. The propagation of a wave is a mechanism that takes place in the medium. The source only initiates the disturbance, the propagation of that disturbance is what the wave is. This process of propagation happens in the medium and thus, the speed of this propagation (i.e., the speed of the wave) depends entirely on the medium. 
A: It is one of Einstein postulates.
Scientific answer on your question based on velocities additional formula:
$$
V^\prime = \frac{V+u}{1+Vu/c^2}
$$
If $V=c$ then $V^\prime = c$.
To understand more deeply you need use Maxwell theory. As convenience of this theory, speed of light is constant in any system frame. This laws was discovered experimentally.
Some physical phenomena have not daily-life analogue, we only try to describe nature around us.
A: The wave-particle duality of photons (or other particles) is I think not relevant for this question.
I won't try to answer the 'why' part of your question, but I will give a description that should help you intuitively grasp how it is possible that light emitted from a moving source appears to travel at the speed of light in any frame of reference.
description 1:

Say there is an observer standing on earth and standing still. A
starship flies past, at a speed of 0.9 c, i.e. 90% of the speed of
light. Define $T_0$ as the moment that the starship is exactly
overhead. At that moment $T_0$ the starship  fires a laser pulse
forward.
Now, at $T_0 + 10$ seconds, in the reference frame of the observer,
the starship will have traveled 9 light seconds, and the laser pulse,
traveling at the speed of light, will have traveled 10 light seconds.
As you mentioned, time passes more slowly for an object traveling at
high speed. Therefore, when according to the observer 10 seconds have
passed, according to the starship pilot only 1 second will have
passed. The distance between the starship and the laser pulse at that
moment is one light second, which is exactly what the pilot would
expect, having fired the laser (in his frame of reference) 1 second
ago.

This description probably does not match entirely with the numbers that the equations of relativity would give you, but it gives an intuition.
Apart from time passing more slowly for a fast moving object, you may know that a fast moving object also becomes smaller in the direction it is traveling in. This allows for a different description of the above situation.
description 2:

We have the same situation as in description 1. A starship flying at
90% of the speed of light passes by the observer at $T_0$ and at that
moment fires a laser pulse forward. After 10 seconds, according
to the observer, the starship has traveled 9 light seconds and the
laser pulse 10, so all is fine.
The starship is nominally 300 meters long, or 1 millionth of a light
second. However because it is traveling very fast, it appears
compressed in the direction of travel, so to the observer the starship
only looks like it is 30 meters long. At $T_0 + 10$ seconds the laser
pulse looks like it has traveled 10 light seconds according to the
observer, or ten million times the nominal length of the starship.
According to the pilot of the starship, the laser pulse has also
traveled ten million times the length of his ship (which does not
appear compressed to him), and therefore the laser pulse has traveled
10 light seconds in 10 seconds, so it is moving at the speed of light
just fine.
The observer will agree with the pilot that the distance between the ship and the laser pulse is ten million times the currently observed length of the starship, but according to the observer that works out to 1 light second. Both the pilot and the observer thus agree that the pulse is traveling at the speed of light relative to them.

The two descriptions don't agree with each other, and if you apply them to more complicated situations they will break down. In the actual physics of relativity both effects play a role (as far as I know). But these descriptions may give you an intuition on how it is possible for observers in different frames of reference to all observer light as traveling at the speed of light.
