Why are mixed models robust to missing (at random) data in the response variable? I have read that mixed effects models are well equipped to handle missing (at random) response data  if estimated using likelihood methods. However, I am yet to find a clear (not overly technical) explanation of how this is achieved. Models without random effects (e.g. general linear models) can be fit using maximum likelihood so are these not well equipped for missing response data too? It always seems to be mixed models that people advocate for missing data, so I'm assuming it must have something to do with the within subject correlations...?
In my reading I came across this
[1]: https://www.uvm.edu/~statdhtx/StatPages/More_Stuff/Mixed-Models-Repeated/Mixed-Models-for-Repeated-Measures1.html
Under the section "Missing data" -  paragraph 3 it says:
But if I can find a way to keep as much data as possible, and if people with low pretest scores are missing at one or more measurement times, the pretest score will essentially serve as a covariate to predict missingness.
I believe the author is intimating about a mixed model approach but do not really understand this sentence and thought that this might be key to my understanding.
Thank you very much!

There are many good references covering the issue that full likelihood methods such as generalized least squares, Markov models, and mixed effects models only require the missing at random assumption, not the missing completely at random assumption.  Missing at random is achieved by not having huge time gaps between measurements, if you can assume that previous measurements before dropout portend the dropout.
The reason that full likelihood methods achieve this is that they record how individual time points for one participant are "plugged into" the likelihood function.  The likelihood "knows" where the gaps are and uses the right weights for the non-missing data.  By knowing how serial data are connected, one gets a more correct analysis when some of the data arent' there.
But note that this is only true if the method you use has a correlation structure that is reasonable.  A random intercepts mixed effects model does not use the ordering of observations within individuals.  It assumes that two responses have the same correlation no matter how far apart in time they were measured.  This is the compound symmetric correlation structure.  This is often unreasonable for serial data.  Instead one usually sees something like an AR(1) correlation structure that has an exponential decay in the correlation as the time gap widens.  This can be modeled using generalized least squares, Markov models, or adding an AR(1) structure on top of a mixed effects model.  If you get the correlation pattern very wrong, the proper connection between multiple measurements within a subject is not used and the result can be unsatisfactory.
