After learning of the Fourier transform in school, I immediately became interested in this for obvious reasons, but never successfully wrapped my mind around its employment usefully, beyond the jargon "wave localization."

The physical significance really arises from a place called the Huygens principle in mechanics, which supports claims about spatial wavefront propagation to state that at each timestep and point on the wavefront, another similar wave propagates, these subwaves are called wavelets.