The signals that we measure in MRI are a combination of signals from all over the object being imaged. It so happens that any signal (even if you simply make one up and draw a squiggle) is composed of a series of sine waves, each with an individual frequency and amplitude. The Fourier transform allows us to work out what those frequencies and amplitudes are. (That is to say, it converts the signal from the time domain into the frequency domain.) Since we encode the signal with magnetic field gradients which make frequency and (rate of change of) phase relate to position, if we can separate out the frequencies we can say where we should plot the amplitudes on the image.
For any image, use of the Fourier transform allows us to manipulate the data in the frequency domain (k-space), which can be easier, and makes things easier to understand, like in the example of a high pass filter.
The answer to this question is explored more fully in the k-space web tutorials.