Given a linear application $F: \mathbb{R} ^3 \rightarrow \mathbb{R} ^3$ and its expression under the form of matrix in standard basis, in order to change its basis to a $b$ basis, vectors of which are given, you would have to multiply the inverse of the change of basis matrix ${[I]}_{b}^{e}$ by the the given linear application in standard basis ${[F]}_{e}^{e}$ and the resulting matrix by the change of basis matrix:
$${[F]}_{b}^{b} = {[I]}_{b}^{e} \cdot {[F]}_{e}^{e} \cdot {[I]}_{e}^{b}$$
and has me very confused.
Isn't this equivalent to finding the coordinates of a vector in a given basis $b$ knowing its vectors and the coordinates of the initial vector in the standard basis?
As here: $\vec{v}_b = {[I]}_b^e \cdot \vec{v}_e$
Why shouldn't you stop after multiplying the linear application with respect to $e$ by the inverse of the change of basis matrix, but rather multiply that matrix by change of basis matrix?
Not 100% sure if that's really what you are asking, but you should pay attention to the fact that you may have different bases in the domain and codomain.
If you only apply the change of basis matrix on the right, you are changing the basis in the domain but not the basis in the codomain.