I understand the formula, but what does it even mean for a Fourier transform to convert from time domain to frequency domain? And how is it doing so? Even, in the field of image processing, why do they say that it converts from spatial to frequency domain?
The Fourier transform decomposes a signal into the weights of orthonormal basis functions: sines and cosines of different frequencies. An analogy would be decomposing a vector in the x-y plane into its x and y components, which are the weights applied to the unit vectors along the x axis and y axis (the orthonormal basis vectors of the x-y plane). Sinusoidal functions can be functions of space as well as time. For image processing you're still decomposing the signal in terms of sines and cosines, the frequencies just correspond to changing in space vs changing in time.