- the Fourier Transform preserves energy (Parceval's theorem, the norm of the transformed function is the same as the norm of the original)
- there exists an inverse transform to get the original function back
- once you grasp that magnitude/phase describe patterns in the function you can gain powerful intuition about the transform and how to use it as an analytical and design tool.
Those first two properties tell us that the transform preserves information, basically it's another way of looking at the same thing to gain more insight without loss. The third is something not harped on enough in engineering courses, and failure to teach it in my mind is one reason so many people thing controls/signal processing is black magic.
A big followup question here is, are there other transforms for which energy is preserved, and there exists an inverse? The answer is yes, there are infinitely many. The third property is more useful, which begs the question, which of those other transforms are useful?
An example of this is a cousin of the Fourier Transform called the Discrete Cosine Transform which is critical in compression, classification, and machine learning (especially contemporary speech recognition). It's not as straightforward as Fourier, since the result isn't as obvious as breaking down the energy into patterns, but rather what it does is break the energy into correlated parts, in other words it preserves the energy while representing it in decorrelated bins. The strongest of those bins is the most important part, which is why compression technology works by taking the DCT and tossing out low magnitude components (it preserves the most important energy) while also showing how it can work for machine learning, where it decomposes the input information into an equivalent but decorrelated representation where inputs aren't shared for different parts of something like a neural net.
There are other equally cool orthogonal transforms, I like the Hilbert transform myself because it can extract really useful info like signal envelopes and be used to make wacky noises, like a frequency shifter.