I've included a picture stating the problem. How do I approach this?
I recognize that $\displaystyle x^2 + y^2$ is the equation of a circle, but that's about all I have.
When you rotate the curve $\displaystyle z = e^{-x^2}$ about the z axis you get the surface $\displaystyle z = e^{-r^2}$ (in cylindrical polar coords). Since $\displaystyle x = r \cos \theta$, $\displaystyle y = r \sin \theta $ then $\displaystyle r^2 = x^2 + y^2$, giving your answer.
$\displaystyle x = r \cos \theta$ and $\displaystyle y = r \sin \theta $ are polar coordinates - an alternate way to describe a point in 2D. Since
$\displaystyle
\sin ^2 \theta + \cos ^2 \theta = 1, \text{then }\;x^2 +y^2 = r^2\left(\cos ^2 \theta + \sin ^2 \theta \right) = r^2
$