More specifically, why do the likelihood ratio tests have asymptotically a $\chi^2$ distribution if the models are nested, but this is no longer the case for the not-nested models? I understand that this follows from the Wilks' theorem, but unfortunately, I don't understand its proof.
Well, I can give a non-rigorous answer from a non-statistician. The Likelihood ratio method relies on the fact that the denominator max likelihood gives a results always at least as good as the numerator max likelihood because the numerator Hypothesis corresponds to a subset of the denominator hypothesis. As a result, ratio is always between 0 and 1.
If you would have non-nested hypothesis (like testing 2 different distributions), likelihood ratio could be > 1 => -1 * log likehood ratio could be < 0 => it is certainly not a chi2 distribution.
In order to undertake hypothesis testing you need to express your research hypothesis as a null and alternative hypothesis. The null hypothesis and alternative hypothesis are statements regarding the differences or effects that occur in the population. You will use your sample to test which statement (i.e., the null hypothesis or alternative hypothesis) is most likely (although technically, you test the evidence against the null hypothesis).
The null hypothesis is essentially the "devil's advocate" position. That is, it assumes that whatever you are trying to prove did not happen (hint: it usually states that something equals zero).
Looking here, we can find this text:
Hypothesis testing is an essential procedure in statistics. A hypothesis test evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. When we say that a finding is statistically significant, it’s thanks to a hypothesis test.
About accepting/rejecting Hypothesis, here, we can find an interesting answer:
Some researchers say that a hypothesis test can have one of two outcomes: you accept the null hypothesis or you reject the null hypothesis. Many statisticians, however, take issue with the notion of "accepting the null hypothesis." Instead, they say: you reject the null hypothesis or you fail to reject the null hypothesis.
Why the distinction between "acceptance" and "failure to reject?" Acceptance implies that the null hypothesis is true. Failure to reject implies that the data are not sufficiently persuasive for us to prefer the alternative hypothesis over the null hypothesis.