Why are edges in spatial images represented as edges in their Fourier transform image?
An edge can be described as a sum of sinusoids of different frequencies with phase all equal to $\pi/2$ (or $-\pi/2$) at the location of the edge.

You might find this paper interesting, it is about how different combinations of sinusoids make up different types of features.

Svetha Venkatesh, Robyn Owens On the classification of image features