How was the Fourier Transform created?
Fourier in his Theorie Analytique de la Chaleur says: "The equations of heat conduction like those of sound or small oscillations for liquids belong to one of the most recently discovered breanches of science which it is important to extend." This passage is quoted at length in Körner's Fourier Analysis.
He goes on to advocate a "calculus" that yields quantitative results for such problems. This involved finding functions that could approximate waves (and other functions), which could serve as solutions of differential equations.
Oscillations and frequency analysis have always been part of this problem.
Fourier's "Memoir on Heat Transmission in Solids" dates to 1807 and is (per Wiki) is considered an important breakthrough. The key insight was that a wide range of functions could be approximated using trigonometric series. Gauss (also per Wikipedia) was the first to use ("discover") the FFT (discrete fast Fourier transform) in studying astronomy in 1805.
While I can't vouch for the Wiki article I think it's a good start.
Edit: Willie Wong's idea that the (continuous) FT was Fourier's invention (and surprisingly that idea of series of this type are not so much Fourier's invention) seems to be supported by an Overflow item here. The author of that post cites a biography of Fourier in support of his claim.
Anon's evanescent comment about orthogonality is certainly part of the answer to this question. Gauss probably gets credit for this idea. He seems to have been first in almost everything else--why not this as well?
According to papers cited in comments, Gauss's version of the FFT was not published in his lifetime.
A short note on the "invention" of the Fourier transform: in Plancherel's "Contribution à l'étude de la représentation d'une fonction arbitraire par les intégrales définies" (1910) Rendiconti del Circolo Matematico di Palermo he wrote (beginning of Chapter 5, p328; translation mine):
Fourier was the first to write down the formula $$ f(s) = \int_0^\infty \cos(x u) \mathrm{d}u \int_0^\infty \cos(tu) f(t) \mathrm{d}t $$ without preoccupying with the issues of convergence of the indicated integrals. After him, other authors worked to establish the validity of the above formula, and to find out the correct conditions under which the formula holds...
Priority claims aside, I think it is quite possible that Fourier in fact did write down the formula for what we call the Fourier transform.
The Fourier Transformation is only an orthogonal projection, with a special scalarproduct.
The main idea is like one can express a point with it's length height and width, to express a function with a linear combination of trigonometric functions.
As it was inventend 1822 I guess it wasn't designed for frequency analysis.
How it was invented? It is motivated from the discrete fourier series, i can upload an example if you like (a plot)