Sorry for the longwinded and vague title. I've included a full picture of the theory (taken from my phenomenal textbook, David Poole's Linear Algebra: A Modern Introduction-Brooks Cole (2014)) below.
Why is this the case? The same elementary matrix operations it takes to reduce C (which is a matrix whose columns are the coordinate vectors under $E$ of the basis vectors of $C$) to I reduces B to the change of basis matrix from $B$ to $C$.
Can someone give impart onto me some intuition as to why this makes sense? Any help is greatly appreciated.
When you apply all the elementary row operations to go from $C$ to $I$, you are effectively left-multiplying the augmented matrix by $C^{-1}$. Then the part where $B$ was before becomes $C^{-1}B$, which is $P_{\mathcal{C}\leftarrow \mathcal{B}}$. (Note that every elementary row operation corresponds to left-multiplication by a certain elementary matrix.)