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In such systems, the multiplicity of microstates decreases as one approaches the maximum internal energy allowed, so their entropy decreases as well.

This sounds trivial but I've never heard of multiplicity of a microstate(I know what multiplicity of a macrostate is) and nothing on the internet points to a definition for this.

Could someone explain to me what this is?

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Almost certainly the multiplicity of a microstate means the number of microstates that are contained within the macrostate that contains the given microstate (and so is the same as the multiplicity of the containing macrostate). If we know that the system is in a macrostate that contains $\Omega$ microstates and if all microstates within that macrostate are equally likely then the entropy of the system is

$$S = k_B \ln \Omega$$

so if the multiplicity $\Omega$ decreases then so does the entropy of the system.

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