Proving the validity of a sequent using Modus Tollens
Modus Tollens is the following rule:

Given $A \to B$ and $\lnot B$, deduce $\lnot A$.

In your case, you are given $p \to (q \lor r)$, and so $A = p$ and $B = q \lor r$. Therefore you need to prove $\lnot B = \lnot (q \lor r)$. This is not the same as $\lnot q \land \lnot r$, though the two are logically equivalent. What you have to do is deduce $\lnot (q \lor r)$ from the premises $\lnot q$ and $\lnot r$.