\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*}