\begin{align} \nabla\left(x_1x_3\hat{e}_1+x_2x_3\hat{e}_2\right)&=x_3\left(\hat{e}_1\otimes\hat{e}_1\right)+x_3\left(\hat{e}_2\otimes\hat{e}_2\right)\\&+\mathbf{x_1\left(\hat e_1\otimes\hat{e}_3\right)+x_2\left(\hat e_2\otimes\hat e_3\right)} \end{align}
Where did the terms in bold come from?
You have to use that $$ \nabla f = \frac{\partial{f}}{x_1}\hat{e}_1+\frac{\partial{f}}{x_2}\hat{e}_2+\frac{\partial{f}}{x_1}\hat{e}_3 $$ Then you get $$ \nabla(x_1 x_3)= x_3\hat{e}_1+x_1\hat{e}_3 $$ and $$ \nabla(x_2 x_3)= x_3\hat{e}_2+x_2\hat{e}_3. $$ The second summands in the last two equations then give you the cross terms $$ \nabla(x_1 x_3 \hat{e}_1+x_2 x_3 \hat{e}_2)= (x_3\hat{e}_1+x_1\hat{e}_3)\otimes\hat{e}_1+(x_3\hat{e}_2+x_2\hat{e}_3)\otimes\hat{e}_2. $$