I read your question a little differently than the other people who have answered. The role of the Fourier transform in physics is as a bi-directional one-to-one mapping between conjugate variables. The most famous example of a conjugate variable pair is position/momentum, but there are many others. Some examples of conjugate variable pairs:
- position and momentum over Planck's constant $[x,p/\hbar]$
- time and energy $[t,E]$
- aperture and the sine of the diffraction angle divided by the wavelength
One of the consequences in quantum mechanics is that the operators (quantum mechanics uses differential operators which are related to the classical variables) do not commute. So, using the position/momentum example; $xp\neq px$. This leads to an uncertainty relation between conjugate variables.
The usual statement of the Heisenberg uncertianty principle is that the more well one knows the position of a quantum particle, the less one can know about the momentum. Mathematically;
However, the more general statement of the uncertainty principle is that the uncertainty between any pair of conjugate variables is given by
This uncertainty relation leads to some rather strange effects in quantum physics. For example, the uncertainty relation between energy and time allow for the violation of energy conservation on very brief timescales, a law which is sacred in classical physics.
If you are looking for more information on the uses of the Fourier transform in physics and engineering with many worked examples, have a look at this free textbook by J. F. James.