The Fourier transform tells you how to express something like an acoustic signal as a combination of something like notes. A trained musician's ear performs something like a Fourier transform when she tells you what notes make up a chord you play on a piano. More generally, your ears perform something like a Fourier transform when you do things like talk to a friend while listening to a television at the same time; you have the capacity to distinguish these different parts of the acoustic signal your ears are getting so you can pay attention to one or the other. This is something like Ohm's acoustic law.
Given the above, it should not be surprising that the Fourier transform is fundamental in signal processing. The Fourier transform is also fundamental in both pure and applied mathematics, as it is very useful for solving certain kinds of partial differential equations. Since acoustic signals are governed by such a differential equation, namely the wave equation, this should not be so surprising either.
This is a vague description. If you want to understand precisely what the Fourier transform is and does, you will have to learn some math. You will need to at least have taken a course on calculus, and having taken courses in differential equations and linear algebra will also be useful. Depending on how thoroughly you want to understand the Fourier transform, it will also be useful to have taken a oourse on real analysis. Once you have these prerequisites there are various sources you can learn from; I like Stein and Shakarchi. If you have some musical background or are otherwise interested in the relationship to music, I like Benson's Music: a Mathematical Offering.