The speed of light in vacuum is a universal constant and important in several branches of Physics. Even the constancy of the speed of light is one of the two postulates of Special Relativity (SR). But my questions are:
Why does the speed of light assumes the particular value 3×108 m/s and not any other different value?
As others said, the digits themselves arise from our choice of units and they are defined only by our convenience.
Also, as @RyderRude explained, you can choose some natural units and then ask about the value in these natural units. The question is then better posed. But then you have a problem that interactions and speed of light are linked together and thus changing the speed of light would probably result in changing the reference of those natural units itself. So as it stands, it is still not well formulated question for scientific research and we must still seek better formulation.
This question can be further reformulated in terms of dimensionless constants of standard model. As they are dimensionless, they do not have any reference frame and are just pure numbers. Standard model of elementary particles contains several such numbers (in fact, 25 of them), like fine structure constant and we may ask wheter there is some reason for its value, or wheter it is random and therefore some different hypothetical universe, which is still based on the same physics principles as our universe, could hypothetically have different value.
The question is open and theories which would predict these dimensionless constants are proposed here and there. The most notorious of such theories is string theory, which contains I think only one dimensionless constant, and its answer to the question is that the 25 dimensionless constants are indeed random and different universe could have different values of these. But we do not know, wheter string theory is true or not, so it is as good answer as any.
The digits of the speed of light are not special. The number that you see ($3*10^8$) only arises in physics because we measure length in meters and time in seconds. But of course, these units are arbitrary. Meters are convenient for human level measurements, and so are seconds.
But to describe the numerical values of distances on larger scales (like inter-planetary distances), we do use other units like Light years (as those distances would have very large numerical values if measured in meters). If we use Lighter year and year respectively as the units of distance and time, then we get $1 light year/year$ as the invariant speed in our universe.
Edit- You're asking why the speed of light is as big as it is. To answer that, we'd first have to link the units of distance and time to some fundamental properties of the universe.
In physics, the fundamental property of the universe that we use to define these units is the speed of light itself. Our definition of a second is the time it takes for light to cover approx $3*10^8m$. If some humans in some alternate universe use this same system to define $m$ and $s$, then they'd inevitably get the same numerical value of the light speed in those units.
So if we want to make sense of the question, we'd first have to define the units of distance and time in terms of some other intrinsic property of the universe, and then ask, if in some alternate universe, the speed of light can have a different numerical value in those fundamental units.
However, I'm not aware of any units of distance and time which are defined in terms of some other fundamental properties of the universe. Even the Planck units are based on the speed of light. Maybe this has something to do with the fact that space and time measurements are not absolute, and hence using the invariant speed to define these units is pretty much the only choice we have.
But let's try to define distance and time units in terms of some other fundamental property. Atoms are pretty fundamental, so let's use the Hydrogen atom. For simplicity, let's assume a non-quantum mechanical world with a Hydrogen atom of a fixed radius and a ball-shaped electron orbiting the nucleus. We could define the unit $m$ as the radius of this hydrogen atom, and the unit $s$ as the time it takes for the electron to orbit completely.
Inevitably, we'd get some fixed numerical value for the light speed in terms of these fundamental units. Our question is now, whether in an alternate universe of Hydrogen atoms, does light speed have the same numerical value in terms of these units?
However, there's a problem with this setup. Would Hydrogen atoms even exist in an alternate universe with a different light speed? I mean...speed of light definitely helps shape up the universe into its current structure. It appears in equations which govern interactions between particles. Who knows what kind of atoms would form or if humans (as we know them) would even exist in a universe with a different light speed. The particle interactions in that universe would probably be very different resulting in a very different universe.
Because of this problem, we're not able to carry over our Hydrogen atom based definition of units from one universe to another.
Take for example, the constant $\pi$. Its value is intrinsically linked with the structure of space on which it is defined. We can't have a different value of $\pi$ for a circle on a flat piece of paper. If we want $\pi$ to have a different value in some universe, the structure of the other universe would have to be completely different.
Similarly, the speed of light is probably linked to the way matter intracts in our universe. If it was any different, universe would be completely different. Maybe atoms wouldn't even form. This is probably the reason it's very hard to even define distance and time units in terms of some other fundamental property of the universe.
As already mentioned in the other answers, the value of the speed of light changes if one changes the (arbitrary) units of length and time. However, the question can be reformulated in a way that does not depend on the units chosen. Let us ask
"Why does the speed of light assumes its particular value and not any other different value, in comparison to other physical constants, such as the electron radius, the electron mass, the fine-structure constant, etc?"
There is no sarisfactory answer to this question. The standard model, which is by far the best description of our universe, does not have an explanation for why the fundamental constants have the value that they have. There is no known reason why the speed of light in comparison with the other fundamental constants. A proposed explanation is the anthropic principle. The values of the fundamental constant seem to be fine-tuned for the existence of life, as we know it. Paraphrasing wikipedia:
"The anthropic principle is the assumption that the reason why we observe the fundamental constants in our universe having the value that they have, is the simple fact that, in order for the universe to be observable at all by a living entity, these constants must have been compatible with the emergence of conscious and sapient life that observes it."
In principle, other parallel universe may have a speed of light which differ from our own universe. In these parallel universe, conscious life may be not possible.
Concluding, I think that the only possible answer to both questions, is the anthropic principle. The discovery or development of theoretical physics in the future may give a better answer to these issues. Before then, the anthropic principle seems to be the more reasonable answer, although not the satisfying one.
EDIT: The value of $c$ affects our universe, and our life in many different ways. For example, the rate of conversion between energy and mass is given by the famous formula $E=m c^2$. That means that a larger value of $c$ will affect the energy radiation of the sun and of other stars. For example a value much higher of $c$ will have the consequence of a much more energetic radiation from the sun, which potentially will kill any life form.
