I'm not sure at what level of formality you want an answer, but an informal and handwavy one would be something like this: an equivalence class of $S$ under $E$ is a subset $C\subseteq S$ such that any two elements of $C$ are related by $E$, and such that there exists ($\exists$) an element in $C$. But Martin-Löf type theories don't really have an $\exists$: they have a $\Sigma$, which, unlike $\exists$, cannot (by construction) forget the witness that witnesses existence. But forgetting the element of $C$ which serves to construct it is what quotienting is all about!
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