The implications of Einstein's first law I'm struggling with the physical meaning/consequence of Einstein's first postulate of Special Relativity, which states that

all physical laws are the same (invariant) in all inertial frames.

Any textbook on special relativity will then continue to say something along the lines:

"Postulate 1 implies that no physical experiment can be used to measure the absolute motion."

I fail to draw this very conclusion myself. I know it has something to do with Newton's second law, $F=ma$, in the sense that Newton's law doesn't apply in a reference frame that is accelerating, but I think I fail to see the bigger picture.
How did they come to the conclusion that we can't measure absolute motion? I understand that it means that we can't determine who is actually moving and who would be standing "absolutely" still.
What is the reasoning? Are we going to assume absolute motion exists, and then derive a contradiction? I understand that if absolute motion were to exist (or at least, to be physically relevant), then we should be able to measure it, but then there would be a "special" reference frame, which is in contradiction with the first postulate. 
This line of reasoning is exactly what I don't understand: why do laws need to be different in order to measure something different/something "special"? Can't it just be that the "absolutely still" reference frame were so special, that we would actually measure different things, while the laws were obeyed?
 A: 
Why do laws need to be different in order to measure something different/something "special"? Can't it just be that the "absolutely still" reference frame were so special, that we would actually measure different things, while the laws were obeyed?

From the first postulate follows that all physical experiments must give the same results in all inertial frames. Imagine you do an experiment in an inertial frame A. Then you move to another inertial frame B and you do an identical experiment. If (from the first postulate) the laws are the same, why would you expect a different measurement? It's exactly the same situation in A and B (forget quantum theory for a moment). I mean, imagine your experiment is measure the period of a pendulum. As physical laws are the same, you would expect the same value in both inertial frames. (Edit: as J. Murray pointed out, I am thinking of performing two identical experiments in two different inertial frames, and the measurements I refer to must be obtained by the comoving observer in each case. I have modified a bit the answer so that it may be better explained now.)
If there's an absolute motion, we can't measure it in any way. This implies that if it exists, it cannot affect us in any (physical) way, because is not measurable! In the practical sense, if we wanted, we could consider that it does not exist in the real world. So we don't care about it.
There is a similar situation when studying the free particle. By definition, a free particle does not interact. If it exists something like a free particle, we couldn't detect it (detection requires interaction). Again, if it exists, it cannot affect us in any physical way, so we can consider that it does not exist in the real world.
A: If the physical laws are invariant under a transformation (relationship between what is measured in one reference frame out of the perspective of another reference frame) that means a reference frame is not unique in accordance with that transformation. The second quotation you posted follows from the first because if there were to exist a frame of reference in which you purport to measure absolute motion one could, due to the invariance of the physical laws under the relevant transformations, find an equally valid reference frame which is not yours and prove that your reference frame was not unique.
The fact that a physical law is not invariant under a transformation is how a special reference frame is defined. There are some transformations which do result in a special reference frame, such is the case with transformations such as translations and boosts where the inertial reference frame condition $d^2x/dt^2=0$ is not obeyed.
A: All physical laws are the same (invariant) in all inertial frames.
Which implies that all physical laws (that we use to predict, and measure physical phenomena) are independent of uniform motion. They are not affected by uniform motion. To Measure something, let's say motion (force), we are going to apply some physical laws but these laws can not sense it in any way and can not tell you that uniform motion is happening! Then conclusion "no physical experiment can measure absolute motion" follows.
