Special Relativity, 2nd Postulate -- Why? As a lowly physics undergrad who has been chewing on this 2nd postulate of special relativity for a year or more, I simply can't wrap my head around reasons why it is true or how Einstein might have been convinced enough to propose this postulate.
Consider Alfred who is riding in a car travelling at 88 m/s with his headlights on and Bernard who is on the side of the road hitch hiking. Why does the light propagating from Alfred's car move at $c$ relative to both Alfred and Bernard, and not at $c$ + 88 m/s relative to Bernard? 
The nifty results of special relativity all kind of hinge on this idea, and asking my professors in class hasnt really yielded an answer much more than "because we have never observed a case otherwise". 
 A: Actually given that the first postulate says that all physical laws are the same in all inertial frames, you could replace the second postulate by the postulate: "Maxwell's equations are the physical laws for electromagnetism".
From Maxwell's laws you can derive that the speed of light in vacuum has a specific, constant value, in SI units $c=1/\sqrt{\epsilon_0\mu_0}$. Now there are three possibilities:


*

*Maxwell's laws are valid only in a specific inertial frame (or rather, in a specific set of inertial frames at rest relative to each other).
That's the essence of the aether hypothesis. It would violate the first postulate. Also, experiments failed to measure that preferred frame.

*Maxwell's laws are not the correct description of electromagnetism (that is, they are valid in no inertial frame).
That option would, of course, have been compatible with the first postulate, but not very likely, given the huge experimental support for Maxwell's equations.

*Maxwell's laws are valid in all inertial frames.
If that is the case, then all of the consequences of Maxwell's equations have to be valid in all inertial frames. One of the consequences of Maxwell's equations is the value of the speed of light in vacuum.
However, it turns out that the only thing from Maxwell's equations you actually need in order to derive special relativity is the constant speed of light. Therefore it makes sense to postulate that directly; that way even if it turned out that Maxwell's theory had to be revised, you don't need to revise relativity as long as the revised theory still predicts a constant speed of light.
A: The answer is simple: Maxwell's equations.
Maxwell published his electromagnetic theory in the 1860s. This generated a huge schism in physics. Maxwell's electromagnetism was in direct conflict with Newtonian mechanics. There is no allowance in Maxwell's electrodynamics for the speed of the emitter or the speed of the receiver. The speed of light is constant per Maxwell's equations. This conflict became even more apparent in 1887 with the Michelson–Morley experiment, which shot down the idea of a luminiferous aether as a carrier of Maxwell's electromagnetic waves.
Many other physicists besides Einstein were trying to attack this conflict at that time. All but Einstein tried to rectify the two theories. The leading contender to Einstein's relativity was the Lorentz ether theory. This solved the problem by handwaving it away. The Lorentz length contraction and time dilation together conspired to hide the ether frame from us. There's no difference mathematically between Lorentz ether theory and Einstein's relativity. Both predict the same outcome for any special relativistic experiment. Yet no one teaches Lorentz ether theory.
Einstein's path was based on the fact Maxwell's equations strongly imply that the speed of light is the same to all observers. Einstein simply took Maxwell's equations at their word. It was such a simple solution to the conflict, and yet it was extremely game changing.
A: The way I think of it, as a non physicist who quite likely has a few things wrong, is as follows: time actually passes slower for Alfred, and thus there is no difference in the speed of light.  
If you say, "Why does time pass slower for Alfred and not Bernard, after all, motion is relative, right!?" Well, this stumped me for a long time as a non-physicist, but ultimately I understand it as: Alfred must have accelerated from an inertial frame that he once shared with Bernard, and thus he is the one who experiences the slowing of time.
If I'm not mistaken, this accounts for the 'ether' parallel drawn in the other reply?
Anyway, as far as how anybody would have been 'convinced' of this, I believe it was due to direct experimental evidence.
A: Einstein did not prove this postulate ; he simply asked "what if it is true?". He had very good reasons for asking that question.
His efforts to answer the question challenged a whole raft of "beliefs" about time and space, none of which were based on proof either ; they were (up until then) assumed true by so-called "common sense" alone. 
He made predictions about how relative speed would dilate time and contract space, and about how acceleration could explain what we call "gravity" and so affect mass and also "bend" light. 
His predictions are supported by experimental results. This is the reason we take his ideas seriously. He always said that his theories would stand or fall on experimental results alone. So far they have stood. 
A: Maxwell's theory had predicted that the speed of light varies with the speed of the observer. Initially (prior to Fitzgerald and Lorentz advancing the ad hoc length contraction hypothesis) the Michelson-Morley experiment was compatible with the assumption that the speed of light varies with the speed of the light source (as predicted by Newton's emission theory of light) and incompatible with the assumption that it is independent of the speed of the source (as predicted by the ether theory).
So in 1905 Einstein's constant-speed-of-light postulate had no justification. It was just a consequence of the Lorentz transforms. The principle of relativity was also a consequence of the Lorentz transforms. Einstein extracted the two consequences, called them "postulates", and deduced the Lorentz transforms from them. He also procrusteanized space and time to fit the new "theory".
