I am a beginner in this quantum-mechanics stuff. I understand the quantum eraser only from an experimental view. So I didn't understand the formalism that describes the quantum eraser. But what does the experiment tells us? Does the photon know that there is somebody watching it? And this is why it behaves in another way? Does the photon also see the future?
No, the photon doesn't see anyone watching it. And the photon doesn't see its future, either. In fact, the photon doesn't exist in any classical sense prior to its observation.
All of its properties - e.g. which slits it could be taking; whether it behaves more as a particle or a wave etc. - are encoded in the wave function until the very moment of the measurement which is why they may always be "changed back" to the previous answers. For example, in quantum eraser, the photon is ordered to behave as a wave again, even though a premature argument could lead a sloppy person to think that the photon has already decided to behave as a particle forever.
When you measure the photon, it is finally possible to think of its properties classically and the wave function allows one to calculate all probabilities that the outcome will be something or something else. In the case of the quantum eraser, we restore the interference pattern. But any attempt to "imagine" that the photon has obtained a classical property at any moment before it was measured would lead to wrong predictions.
It is always essential to appreciate that the photon always behaves according to the laws of quantum mechanics and we're never allowed to approximate it by any classical intuition because the classical intuition fails. This strict requirement that classical mechanics is wrong may only be partly circumvented after the photon is actually detected (because then it interacts with a classical object that quickly decoheres) - but not earlier than that. In other words, quantum mechanics always holds: that's the main lesson of this experiment (and many others).
Sb1 says that it was remarkable that the experiment behaved as Scully and Druhl predicted. I disagree with this wording. The prediction could have been made by any father of quantum mechanics - no new physics was used whatsoever and they could predict the behavior of any setup of this kind. It could have been remarkable in the 1920s but after the 1920s, all such experiments were mundane physics.
"Quantum eraser" was first proposed by M. Scully and K. Druhl. In order to understand it, we must know the famous double slit experiment first which I suppose you already know. You must be aware that in the double slit experiment if photons are emitted one at a time an interference pattern forms at the detector screen. As soon as you try to observe which path the photon follows that is which slit it passes through, left or right slit, the interference disappears. That means a knowledge of "which path" information destroys the wave like character of a photon and hence no interference possible. But in 1982, Scully and Druhl suggested a stunning modification of the experiment. They proposed the following on the basis of their quantum mechanical calculation.
Suppose a tagging device is attached by which we can know the "which path" information of the photon. Now if, just before the photon hits the detector screen, we eliminate the possibility of our knowledge of the "which path" information by erasing the mark registered by the tagging device, both possibilities that is the photon passed through the left slit and photon passed through the right slit should come back into play. Both histories should come back once again and interference pattern should reemarge. As if we are kind of shaping the past (warning: it is by no means that future is affecting the past).
Experiment carried out by Raymond Chiao, Paul Kwiat and A. Steinberg. Remarkably it worked just as scully and Druhl predicted. Interference pattern indeed reemarged.
All the other explanations are wrong. The light does not interact with itself anymore because the light is polarized. Only waves polarized in the same direction can interfere with each other. It's just another dimension. Do a "measurement" that doesn't use interference and it won't work.
A photon has no memory, but the photon/electromagnetic wave "knows" its polarization and only reacts (interferes) with waves polarized the same way.
There is no waveform that collapses by observation, the only thing that happens is that the measurement is not actually a measurement but a change of the system. A pure measurement would only measure and not disturb the process.