What Is the Fourier Transform?

The FT, as are many other transforms, are 1-1, so, in theory, there's no information lost or gained. In many real world conditions, looking at a function in frequency space greatly reduces the problem. Why? Pet theory: because many functions that look complex are actually composed of simpler building in the transformed space.

Take the sound wave of a fly and it looks horribly complex. Pump it through the FT and you find a main driver of the wings beating at a single frequency. Take the sum of two sine waves and it looks a mess. Take the FT and you see the signal neatly broken into two peaks. Etc.

The use of the FT (or DCT or whatever) for JPEG, MP3 or the like, is basically exploiting this fact by noticing the signal response for human hearing and seeing it's not uniform, and so can be "compressed" by throwing away frequencies we don't care about.

The "magic" of the FT, and other transforms, isn't so much that it transforms the signal into a set of orthogonal basis but that many signals we care about are actually formed from a small set of these signals, allowing the FT and cousins to notice and separate them out more easily.

And there's another good reason why so many real-world signals are sparse (as you say) in the FT domain in particular: because so many real-world systems involve periodic motion (rotating motors, fly's wings as you noted, etc). When the system is periodic, the FT will compress the signals very effectively because every signal has to be harmonic of the fundamental frequency.

Well, stable systems are can either be stationary or oscillatory. If the world didn't contain so many stable systems, or equivalently if the laws of physics didn't allow so, then likely life would not have existed. All life is complex chemical structures, and they require stability to function. Ergo, by this anthropic argument there must be many oscillatory systems.