How does a 2D Fourier Transform of an image work?
A 2D Fourier transform is performed by first doing a 1D Fourier transform on each row of the image, then taking the result and doing a 1D Fourier transform on each column.  Or vice versa; it doesn't matter.

Just as a 1D Fourier transform allows you to decompose a function into a sum of (1D) sine waves at various frequencies, a 2D Fourier transform decomposes a function as a sum of 2D sine waves.  These waves can have different frequencies along the x and y axes.  They generically have the form:

$$ \exp \bigl(i \cdot (k_x x + k_y y) \bigr) $$

where $k_x$ and $k_y$ are the frequencies along the $x$ and $y$ axes. These two values form a vector called the wave-vector.  In the spatial domain, the wave is oriented along the $(k_x, k_y)$ vector with a frequency along its axis of $\sqrt{k_x^2 + k_y^2}$.

Just as for the 1D Fourier transform, there exist both discrete and continuous versions.  The result of a discrete 2D Fourier transform is a matrix of complex amplitudes for a set of discrete $(k_x, k_y)$ values.  This is commonly visualized (like in the paper you linked to) as an image where the pixel at coordinates $(k_x, k_y)$ represents the amplitude of that wave-vector.

So, an annular shape in a 2D Fourier transform indicates rotational invariance of the distribution of frequencies (i.e. just as much amplitude for waves in every direction), with a narrow range of magnitudes (from the inside of the annulus to the outside).  In other words, the paper is using the Fourier transform to demonstrate that their noise is reasonably isotropic and band-limited.