\begin{align*}
\left(\hat{I}\otimes{}\left|0\right>\left<1\right|\right)
\left(\left|0\right>\otimes{}\left|0\right>+\left|1\right>\otimes{}\left|0\right>\right) & = 
\left(\hat{I}\otimes{}\left|0\right>\left<1\right|\right)\left(\left|0\right>\otimes{}\left|0\right>\right)
+ 
\left(\hat{I}\otimes{}\left|0\right> \left<1\right|\right)\left(\left|1\right>\otimes{}\left|0\right>\right)\\
& = \left(\hat{I}\left|0\right>\otimes{}\left|0\right>\left<1\right|\left.0\right> \right)
+
\left(\hat{I}\left|1\right>\otimes{}\left|0\right>\left<1\right|\left.0\right> \right) \\
& = \left|0\right>\otimes{}\left|0\right>0+\left|1\right>\otimes{}\left|0\right>0 \\
& = 0
\end{align*}