Actually, this is not a consequence of Zariski's Main Theorem, but rather of the Theorem of Formal Functions (which is used to prove Zariski's Theorem, too).

In fact, the Theorem of Formal Functions implies the following result, see [Hartshorne, Algebraic Geometry, Corollary 11.2 page 279].

Proposition. Let $f \colon X \to Y$ be a projective morphism of noetherian schemes, and let $r= \textrm{max} \{\dim X_y | y \in Y \}$. Then $R^qf_* (\mathscr{F})=0$ for all $q >r$ and for all coherent sheaves $\mathscr{F}$ on $X$.

Now by projection formula we have $$R^qf_{0*}(T_{X_0/Y_0}\otimes f_{0}^{*}I))=R^qf_{0*}(T_{X_0/Y_0}) \otimes I,$$ so you can apply the previous proposition with $r=1$ and $\mathscr{F}=T_{X_0/Y_0}$ in order to get the desired vanishing.

Probably Illusie wrote "Zariski's Main Theorem", but he intended the Theorem of Formal Functions (which is the key result needed in the modern proof of Zariski's Theorem).

In fact, the Theorem of Formal Functions implies the following result, see [Hartshorne, Algebraic Geometry, Corollary 11.2 page 279].

Proposition. Let $f \colon X \to Y$ be a projective morphism of noetherian schemes, and let $r= \textrm{max} \{\dim X_y | y \in Y \}$. Then $R^qf_* (\mathscr{F})=0$ for all $q >r$ and for all coherent sheaves $\mathscr{F}$ on $X$.

Now by projection formula we have $$R^qf_{0*}(T_{X_0/Y_0}\otimes f_{0}^{*}I))=R^qf_{0*}(T_{X_0/Y_0}) \otimes I,$$ so you can apply the previous proposition with $r=1$ and $\mathscr{F}=T_{X_0/Y_0}$ in order to get the desired vanishing.