Consider the random point $(X,Y)$ in $\mathbb{R}^2$. The ratio $X/Y$ tells us what angle the segment from $(0,0)$ to $(X,Y)$ makes with the $x$-axis, while $X^2+Y^2$ tells us how far $(X,Y)$ is from $(0,0)$.
The distribution of $(X,Y)$ is symmetric under rotations, so the distribution of the angle $\Theta$ is uniform and independent of the radius $R=\sqrt{X^2+Y^2}$.
This intuitive explanation can be made more rigorous by converting to polar coordinates.
The joint density of two independent standard normals $(X,Y)$ is $f(x,y)={1\over 2\pi} \exp(-(x^2+y^2)/2).$ Converting to polar coordinates we get the joint density of $(R,\Theta)$ as $g(r,\theta)={r\over 2\pi}\exp(-r^2/2)={1\over 2\pi}\cdot r\exp(-r^2/2).$ This product form of $g(r,\theta)$ shows that $\Theta$ and $R$ are independent.
I don't understand how to map it backwards, that is get the gaussian random variable from the radium and angle pdf ?