This is quite thoroughly discussed in this review: Random Matrix Theory and Chiral Symmetry in QCD by Verbaarschot and Wettig.

The gist is that in the deep infrared, the chiral random matrix model belongs to the same universality class as QCD does, so one can hope to obtain all the universal results of infrared QCD. The review mentions one such application is the derivation/reproduction of Leutwyler-Smilga sum rules.

My lack of background on random matrix theory prevents me from reading far into that review, I hope I can add more in the future.