Why A Fourier transform decomposes a waveform sampled over time into the set of sinusoidal components that make it up, giving amplitude (and phase) versus frequency. The Fast Fourier Transform is simply an efficient algorithm for computing that discrete transform, which is what lets an SDR build a waterfall or panadapter display from a stream of samples. So its job is to take time-domain samples in and produce a frequency-domain spectrum out.
Watch out Going from the frequency domain back to the time domain is the inverse FFT, a separate operation; converting between analog and digital is the job of the ADC and DAC, which feed the FFT rather than being it.
FFT = time in, frequency out. Add an 'I' for inverse to go back the other way.