\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(\left|0\right>\otimes{}\left|0\right>+\left|1\right>\otimes{}\left|0\right>\right)\otimes{}
\left[
\left|0\right>\left<1\right|
\left(\left|0\right>\otimes{}\left|0\right>+\left|1\right>\otimes{}\left|0\right>\right)
\right]
\end{align*}