A rather simple response is to differentiate the characteristic polynomial and use your interpretation of the determinant.
$$det(I-tf) = {t^n}det(\frac{1}{t}I-f) = (-t)^ndet(f-\frac{1}{t}I)= {(-t)^n}\chi(f)(1/t)$$
So if we let $\chi(f)(t) = \Sigma_{i=0}^n a_it^i$, then ${(-t)^n}\chi(f)(1/t) = (-1)^n\Sigma_{i=0}^n a_it^{n-i}$
But $I-tf$ is the path through the identity matrix, and $Det(A)$ measures volume distortion of the linear transformation $A$.
$$det(I-tf)^{(k)}(t=0) = (-1)^nk!a_{n-k}$$
and a change of variables ($t\longmapsto -t$) gives (and superscript $(k)$ indicates $k$-th derivative)
$$det(I+tf)^{(k)}(t=0) = (-1)^{n+k+1}k!a_{n-k}$$
So the coefficients of the characteristic polynomial are measuring the various derivatives of the volume distortion, as you perturb the identity transformation in the direction of $f$.
$$a_k = -\frac{det(I+tf)^{(n-k)}(t=0)}{(n-k)!}$$