However I suspect that I am not fully grasping the essence of the Satorra-Bentler Scaled Chi-Square?

you are indeed not quite grasping it.

All I know is that for example, during some MGCFA (Multi-group Confirmatory Factor Analysis) to test for model invariance they do use the Satorra-Bentler Scaled Chi-Square and test for differences.

you don't only need it when doing multi-group CFA. you can use the satorra-bentler correction for any kind of factor analyses (even traditional EFA if you fit it via maximum likelihood)

Is there anyone who could explain me with practical examples what is the Satorra-Bentler Scaled Chi-Square?

very quick, step-by-step explanation:

- most traditional SEM modeling assumes your data comes from a multivariate normal distribution

- if your data does not come from a multivariate normal distribution some of its estimates are not optimal. more specifically excess kurtosis makes the standard errors of your parameter estimates to be too small (so you don't have the correct 5% type 1 error rate for factor loadings, factor correlations, etc.) and the chi-square test of fit is too big (so you over-reject correctly specified models).

- a way to deal with this is by fixing the model chi-square using some sort of correction so it returns to its to its nominal type 1 error rate. Albert Satorra and Peter Bentler came up with a way (which i'm conveniently not explaining because i'm unsure of the level of mathematical expertise that you have) and the literature has found that it works pretty well, so now we have that to work with.

the way most people use it is the test for multivariate kurtosis using Mardia's measure of Multivariate Kurtosis and if they find it significant, then they apply the Satorra-Bentler correction.

i found this article to give a very nice (and i think readable) explanation of the effects that non-normality (e.g. non-0 skewness and excess kurtosis) have on SEM. as you can read on the introduction, it is meant for introductory and didactic purposes so i'm hoping you'll find it useful.

it talks about the Satorra-Bentler chi square in the 3rd section of the article.

Yuan, K. H., Bentler, P. M., & Zhang, W. (2005). The Effect of Skewness and Kurtosis on Mean and Covariance Structure Analysis The Univariate Case and Its Multivariate Implication. *Sociological Methods & Research, 34(2)*, 240-258.