For example I would love to find an article that starts with "let's make a wav file smaller". And then somewhere in the middle it just says "and here we will use FT to achieve X".
Honestly though, the Fourier transform is useful anywhere that it's easy to think in terms of frequency instead of time. 90% of use cases in the real world come about from trying to do something that's that's easy to express in terms of signal frequency and difficult to express in terms of signal time/space. You can just use the Fourier transform to convert between them efficiently.
To give a morbid, but non-obvious example, I once did some work with a charity that was concerned with migrant deaths. They wanted to figure out where the migrants most at risk were working to approach the farmers hiring them directly. We got a dataset of when migrant bodies were found and did an FFT (among other processing) to find the periodicity of the crops that they were coming to harvest. There's only a few major crops and they tend to have distinct growing periods, so this is enough information to pinpoint certain crops like strawberries.
Mainly is tool for to obtain information that will feed latter other algorithms. Also can be used as a signal filter with the Inverse FT.
If this does help to understand, it is homologous to use several band pass filters repeatedly for trying to obtain the root signals, but without adjustments requirements, highly faster to compute, simpler, and with better result.
So your question is, What are the uses of knowing the main elements that make up a signal?
That information is useful for analysis of sounds or images, for to detect the presence of elements composing the signal out of the expected range.
Also for pattern detection, as you may give the data a signal form of your own, and analyze the frequencies peaks for example.
For compression, the above two, as knowing the main cyclic elements and their intensity allows to determine if some ones may be latter omitted (for loss compression), or what elements latter should be replaced as parameters that the decompression algorithm will use for to reconstruct the signal.
Also it is important what said the other answer. It does not give information about the time/space moment in with is produced the element, it only tells the element is present (the frequency in the signal).
Note that the spectrograms in [0] use what's called a short time fourier transform[1]. Basically, split your song into windows (e.g. 1 second each), and do a fourier transform on each window. This lets you get a picture of the spectrum over time.
[0] https://sound.stackexchange.com/questions/38709/320kbps-mp3-...
[1] https://en.wikipedia.org/wiki/Short-time_Fourier_transform
You take a signal, take its fourier transform, and then cut off the frequencies above some threshold you don't care about (I'm ignoring certain complicating factors for simplicity). Let's say your original signal has frequencies up to 22KHz. If you're only interested in moderate fidelity human voices talking, maybe you cut off everything below 100 Hz and above 3.4 KHz. You just toss that information. Then on the receiving end, you do an inverse DFT and reconstruct the signal.
JPG and MP3 do something like this, along with a pile of other tricks.