CERN and electroweak interaction I know that the EM and the weak force were described in the same mathematical formalization by GSW and that it was predicted that they would appear as one at energies of 200 GeV. I know that the $W^+$, $W^-$ and $Z^0$ were discovered in 1982. So what ? The carriers of the weak force were discovered. How can this mean that the EM and the weak force were unified at the moments of these CERN collisions ?
Kindly explain this simply, if you can, because I want to explain to my high-school pupils and I think that answers involving theoretical representations SU(2)$\times$U(1) and symmetry breakings etc. which were proven by CERN to be correct actually only show that I hide behind mathematics and that I haven't really understood.
And furthermore, why does the weak force only affect neutrons and protons ?
Thanking you in advance, I am waiting for your help
 A: Regarding unification without mathematics, I think it's best to dial it back to before spontaneous symmetry breaking. There was a EM-like force mediated by the $A$-boson, and a weak force, which was mediated by a triplet of bosons: $W^{\pm}$ and the $W^0$. They were all massless.
Along comes the Higgs mechanism, and the $A$ and $W^0$ get mixed into the photon ($\gamma$) and the $Z$. The $W$ and $Z$ acquire mass. That means light is partially composed of $W^0$s. I think that would have blown Maxwell's mind.
CERN in 1982 was of course, post spontaneous symmetry breaking. What they did was create the $W$ and $Z$ bosons, at the expected masses based on electroweak theory. There was nothing special about the reactions that meant there was unification or not.
For a simpler example, look at SLAC/SLC. At the time this was an
electron-positron collider designed to make $Z$s. One reaction is:
$$ e^+e^- \rightarrow e^+e^- $$
Of the many ways this can happen, two are:
$$ e^+e^- \rightarrow \gamma \rightarrow e^+e^- $$
$$ e^+e^- \rightarrow Z^0 \rightarrow e^+e^- $$
where the intermediate particle is virtual. At low energy, the electromagnetic reaction is completely dominant, the weak force is just too weak. As the collision energy is raised, the EM channel proceeds at a very predictable pace. As you approach the $Z$ mass, the cross section goes way up. Hugely. This is because of resonant production via the weak channel.
Also: the EM coupling is no longer $\alpha \approx 1/137 $, it's more like $1/128$ because of the running coupling. Meanwhile the weak force is no longer weak, it's coupling is something similar.
So: the photon and $Z$ have similar coupling and under go the same reaction. They're so similar, that these 2 channels can interfere quantum mechanically, leading to parity violation for polarized leptons.
Had the experiment been done pre High Mechanism, I suppose one would consider:
$$ e^+e^- \rightarrow A \rightarrow e^+e^- $$
$$ e^+e^- \rightarrow W^0 \rightarrow e^+e^- $$
but that's just not something people talk about.
In summary: The unification is not something we recreate in the lab. It is theoretical, but it has implications for what we measure in the lab today.
It very much like the unification of the electric field and the magnetic field: the special theory of relativity taught use they are 2 parts of the same thing, an antisymmetric 4-tensor field. Likewise, EM and the Weak interaction are 2 parts of a more complicated thing called the electroweak interaction.
A: The unification of the forces in electroweak theory was proposed because of mathematical beauty essentially, it just seemed right to have a unified gauge multiplet of the spin 1 bosons that interacted with the leptons. You can see the original paper from Weinberg here, but the idea goes back further, at least to Julian Schwinger. I'm not an expert on QFT, so you'll have to wait for someone else to explain why that seemed so natural to physicists during that time.
I can tell from your question that you're also confused about another point. The forces were not unified "at CERN" when they found the W and Z... They are always unified in that the simplest description of the dynamics of these fields requires you to consider them together as a package.
Just like electricity and magnetism need to be thought about a single unified phenomena to get the simplest picture, the same can be said about the electroweak theory. The Ws, the Z, and the photon are mixtures of 4 other sibling particles called the As and the Bs as well as some components of the Higgs field. These As and Bs are massless, just like the photon. When the universe got cold enough, certain superpositions of these As and Bs and the Higgs got locked in to reality via electroweak symmetry breaking. The end result is that we have a massless neutral photon that couples to electric charge, and three massive vector bosons, the W+, W-, and Z.
That we found the Ws and Z experimentally, and that their mass ratio is consistent with what we expect given their coupling strengths to other particles is what tells us the unified theory is correct. I.e. when we use our mathematical picture of a unified electroweak theory, we get some predictions about which particles exist and how they interact with matter; those predictions are correct.
A: Unfortunately there are two issues here, which are often ignored in popular descriptions.
The first is that "electroweak unification" is not really unification at all, in that above the electroweak symmetry breaking scale, the weak and electromagnetic forces are not different aspects of the same force. The gauge group remains a product, $SU(2) \times U(1)$, and hence there are two totally independent gauge fields. Contrast this with grand unification, where above the symmetry breaking scale the gauge group is simple, such as $SU(5)$. In that case we really do have only one force. (I wrote an answer with a little more detail about this here.)
The second is that electroweak symmetry is not unbroken in LHC collisions. The electroweak symmetry breaking scale is conceptually a temperature, not an energy. If you live inside a giant block of ice, speeding up two molecules of it doesn't turn all the ice to water.
It's possible to explain this stuff fully and correctly, but it's probably going to take longer than you can afford for your students. In my opinion, the best approach is to just get them drawing Feynman diagrams, and save the big words for later. A great resource for that is here.
A: This question is conceptually related to this one, and it will help if you read my answer there. It is the difference between statistically determined and kinetically determined quantities for single interactions. For water, look at the curves for 0 temperature and 100C. The individual molecules have a probability spectrum for their velocity.

When the temperature, which is defined by the average molecular kinetic energy reaches 100 C a phase transition happens, dependent on the total sample and not on the individual molecules.
It is more complicated with symmetry breaking in quantum mechanics, but conceptually similar. The elementary particles  in the table  are described as quantum mechanical fields covering all of space, on which quantum mechanical operators act to generate an electron (for example) with a creation operator, and propagate it on the electron field . It is the symmetry breaking of the fields that is characterized by the scale of 200GeV, not of the individual particles. This can only happen when, in cosmology, the temperature of the universe, i.e. the average kinetic energy of all the particles in it, are at 200 GeV, not just the kinetic energy of single particle interactions.
This happens at ~ $10^{-12}$ seconds after the Big Bang in the  cosmological timeline:

Hope this helps.
Edit after comment:
To address your specific questions:

I know that the W+, W- and Z0 were discovered in 1982. So what ? The carriers of the weak force were discovered. How can this mean that the EM and the weak force were unified at the moments of these CERN collisions ?

The discovery validated the electroweak theory of the standard model, 

In particle physics, the electroweak interaction is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different aspects of the same force. Above the unification energy, on the order of 246 GeV, they would merge into a single electroweak force. Thus, if the universe is hot enough (approximately $10^{15}$ K, a temperature exceeded until shortly after the Big Bang), then the electromagnetic force and weak force merge into a combined electroweak force. During the quark epoch, the electroweak force split into the electromagnetic and weak force. 

and the final validation was the recent discovery of the Higgs meson. The theory is a quantum field theory, but to be true, there have to be particles generated on the fields when there is enough energy. If they had not been discovered , the theory would have been falsified. The symmetry breaking mechanism is mathematically complicated.
I tried to give an intuitive understanding above. As in classical phase transitions, from liquid to gas, the individual molecules can have very high kinetic energies,  but the collective transition happens when the average of the ensemble reaches a temperature corresponding to an average velocity, so the fields describing elementary particle interactions have to be spontaneously broken,at some time in our universe not the individual interactions as seen in the events measured. What we see in the masses of W and Z is proof that the theory works and describes data.
The weak force affects most of the particles in the standard model table, not just protons and neutrons. It was first studied in neutron decays, but there is a large number of data that involves most of the table, as stated in the comments to your question.
