What is the link between interpretability hierarchy and consistency strength

Question

I am trying to understand this definition https://plato.stanford.edu/entries/independence-large-cardinals/#IntHie of Interpretability Hierarchy and how it relates to the concept of Consistency Strength. For example, I have found on some lecture notes that $ZFC+CH \equiv ZFC \equiv ZFC + \neg CH $ but this claim comes without proof. Now, I know that $Con(ZFC) \leftrightarrow Con(ZFC + CH) $ and that $ Con(ZFC) \leftrightarrow Con(ZFC + \neg CH) $. Moreover, the two notions seem to be related somehow, so I wonder what the exact connection is.

Answer 1

Belatedly turning my comment into an answer:

Consistency strength calculations are generally proved as corollaries of sharper results. E.g. the proof that $\mathsf{ZFC}$ and $\mathsf{ZFC+\neg CH}$ are equiconsistent (relative to $\mathsf{PA}$, say) follows by exhibiting an interpretation of the latter in the former.

The point is that if $T$ interprets $S$ and $S$ is inconsistent, then $T$ must also be inconsistent; contrapositively, if $T$ is consistent, then so is $S$. So interpretability results yield relative consistency results, and the former are indeed the chief means of attaining the latter.

In a sense, interpretability is a better notion than relative consistency. Remember that relative consistency is both dependent on a choice of "base theory" and somewhat capricious; for example, if we take as our base theory $\mathsf{PA}$, then the linear order axioms and the group axioms are boringly equiconsistent since $\mathsf{PA}$ proves that each is consistent outright.