The notion of invertibility and isomorphism are telling us when two vector spaces are “essentially the same”. To get a better sense of why this is true we consider the following:

If $V$ is a finite dimensional vector space with $dim(V)=n$, then what this theorem is telling us is that $$V\cong \mathbb{F}^n$$ If we are trying to solve a problem about $V$, then we can try to translate that problem to a problem about $\mathbb{F}^n$ using an isomorphism $T:V\to\mathbb{F}^n$, solve the problem using the techniques we’ve learned in $\mathbb{F}^n$ and then transfer the solution back to V using $T^{-1}$