Reason of Fast Fourier and Inverse Fast Fourier
FFT and IFFT are algorithms that implement the (inverse) discrete Fourier transform (DFT). These transforms convert a signal into another representation, namely the frequency domain, and back. This other representation allows us to analyze certain properties of the signal and also to change these properites which would be hard to do in the original representation.

Consider a digital audio signal: it's a sequence of samples that represent amplitude over time. This representation is useful to study the temporal properties of the signal and to play it back to make it audible. We can hear that the audio signal contains low and high sounds (frequencies) but from looking at the signal it's impossible to tell what frequencies exactly. After applying the DFT to the signal we can analyse what frequencies it contains and how strong they are. We can even amplifiy or attenuate certain frequencies and then apply the inverse DFT (IDFT) to make the signal audible again. In this way we have built a filter. The application of IDFT is necessary in this case, because the frequency domain representation of the audio signal is useless for playback.

Image processing deals with two-dimensional signals and therefore two-dimensional (I)DFT. The concept of frequencies with images is not so illustrative as with audio signals. Edges, i. e. sudden changes of intensity are represented by high frequencies whereas smooth gradients are represented by low frequencies. This allows us to do edge detection, for example. If high frequencies are attenuated this will result in a smoothed image after application of IDFT. Again, for just looking at the image, the frequency domain representation is useless but it allows analysis and filtering.