Mathematically, the Fourier transform is "simply" a way of representing time signals in a certain orthogonal vectorial basis. Vectors in an ordinary sense, e.g. a displacement vector on Earth's surface can also be represented in several orthogonal bases: one basis could, for example, be two vectors pointing North and East; another could be a vector pointing along a certain road and one perpendicular to it. There is nothing inherently special about any of these bases, one could draw maps according to any of these two or many other conventions. (Orthogonal basis vectors are not even necessary, only convenient.)
The interesting thing about time-dependent signals (or any "pretty" function, really) is that they live in an infinite-dimensional vector space, which is hard to imagine; but (besides some important technicalities) the math works mostly the same way: signals as infinite-dimensional vectors can be represented in a lot of bases. One representation is the Fourier transform, where the basis vectors are harmonic functions. The "map" showing the shape of a signal as a combination of infinitely many harmonic functions -- i.e. the frequency domain -- is just as real as any other map with different basis vectors, e.g. the Walsh–Hadamard transform mentioned in the article. And, crucially, the original time-domain representation is also just one map showing us the signal, though it is often the most natural to us.
The interesting thing about time-dependent signals (or any "pretty" function, really) is that they live in an infinite-dimensional vector space, which is hard to imagine; but (besides some important technicalities) the math works mostly the same way: signals as infinite-dimensional vectors can be represented in a lot of bases. One representation is the Fourier transform, where the basis vectors are harmonic functions. The "map" showing the shape of a signal as a combination of infinitely many harmonic functions -- i.e. the frequency domain -- is just as real as any other map with different basis vectors, e.g. the Walsh–Hadamard transform mentioned in the article. And, crucially, the original time-domain representation is also just one map showing us the signal, though it is often the most natural to us.