@trebor So you can think of it as still classifying something like torsors, but now appropriately higher-dimensional. The relevant word here is "gerbe" (I think @buchholtz has a talk on such things in HoTT).
The fact that higher EM spaces classify n-gerbes is one of the results in higher topos theory (Theorem 7.2.2.26). It's also one of the motivations given in the preface, which is worth reading
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