How then, would one be able to simulate say CHSH, which produces fundamentally quantum probabilities that cannot be explained locally/classically? Am I misinterpreting the meaning of simulate?
Quantum phenomena cannot be "explained classically" only when locality is taken into consideration.
In other words, classical phenomena cannot reproduce (some types of) quantum correlations provided that we don't allow for certain types of correlations.
As a concrete example, consider a standard CHSH scenario. We can compute the outcome probability distributions for each measurement setting (it's what you do when you study the protocol), therefore you can trivially write some code to "simulate" the results of an experiment, meaning to draw a possible sequence of measurement outcomes you would find in an experiment. But this is clearly not the same as observing nonlocality with a classical computer: you would just be crunching some numbers that you know, in some situations, can be interpreted as markers of nonclassical correlations.
Put in another way, you can always sample from an arbitrary probability distribution $p(ab|xy)$. Whether such a distribution is "nonclassical" is only meaningful in relation to some imposed restriction (e.g. defining "classical" when it can be written as $p(ab|xy)=\sum_\lambda p_\lambda p_\lambda(a|x) p_\lambda(b|y)$). When you simulate such a distribution on a computer, you don't need to respect such restrictions, so there is no problem.
In general, how could a classical computer simulate quantum phenomena that cannot be explained classically
Aside from locality constraints, such as those described above, quantum mechanics does not predict output probability distributions that are incompatible with classical physics. The difference is in how those outputs can be obtained: quantum mechanic can produce output probability distributions in a radically different way than what classical physics allows for, and in some cases these new behaviours are more efficient.