When an input signal is modulated such that it is multiplied by Acos2(pi)fc its frequency is shifted by fc, which is the frequency of carrier signal. This is called frequency shifting.
The Fourier Transform is extensively used in the field of Signal Processing. In fact, the Fourier Transform is probably the most important tool for analyzing signals in that entire field. So what exactly is signal processing? I'll try to give a high level overview. Plz watch complete ideo.