Image processing and the Fourier Transform
At a conceptual level, the Fourier Transform tells you what is happening in the image in terms the frequencies of those sinusoids. For example, if you have a picture of a plain wall, the values of the pixels change very little as you go from left to right or from top to bottom.  In the frequency domain that means that your image contains low frequencies, but no high frequencies.

On the other hand, if you have a picture of a picket fence, then the values of the pixels change all the time as you go from left to right.  So in Fourier domain you have high frequencies in the X direction, but not in the Y direction.

Finally, if you have a picture of a checkerboard, then the pixel values change a lot in both directions.  Thus the Fourier transform of the image will have high frequencies in both X and Y.

Because the Fourier transform tells you what is happening in your image, it is often convenient to describe image processing operations in terms of what they do to the frequencies contained in the image.  For example, eliminating high frequencies blurs the image.  Eliminating low frequencies gives you edges.  And enhancing high frequencies while keeping the low frequencies sharpens the image.

FFT is used extensively in image processing and computer vision. For example, convolution, a fundamental image processing operation, can be done much faster by using the FFT.  The Wiener filter, used for image deblurring, is defined in therms of the Fourier transform.  But more importantly, even when the Fourier transform is not used directly, it provides a very useful framework for reasoning about the image processing operations. 

Steve Eddins, one of the authors of "Digital Image Processing with MATLAB", has a whole series of blog posts on the Fourier transform and how it is used in image processing.