By the construction of derived functors, $\mathrm{Tor}(I,A)$ is the homology of the complex
$$0 \to E_{n-1} \otimes A \to \cdots \to E_1 \otimes A \to E_0 \otimes A.$$
(All tensor products are $\otimes_S$.) In the particular case that $A = S/\mathfrak{m} =:k$, we have
$E_p \otimes k= \oplus_j k(-a_{pj})$, so $\mathrm{Tor}_p(I, k)$ is the homology of the complex of $k$-vector spaces defined by taking the matrices in your original complex and reducing all of their elements modulo $\mathfrak{m}$. The assumption that your complex is minimal is equivalent to saying that all of the maps are $0$ modulo $m$, so you are just taking the homology of a complex where all maps are zero, and you get
$$\mathrm{Tor}_p(I, k) \cong \oplus_jk(-a_{pj})$$
as a graded $k$-vector space. Now use the definition of CM regularity.