Your sentence describes what the Fourier transform is, whereas the sentence in the article tries to describe how the Fourier transform does it.

What the article does is translate the mathematical notation to a sentence in English, attempting to retain all the information in the mathematical notation. It can be helpful if you are trying to visualize what each component of the mathematical notation is doing, but if you separate the sentence from the equation, it is not very helpful. For example, how would you interpret "spin your signal around a circle"?

It's confusing because it explains "how." Your sentence explains "what," which misses the original point.

If you understand the "what" deeply, then the "how" is trivial. The decomposition is in the sense of an orthogonal basis, and so to compute the transform you just do a dot product in the relevant vector space.

The key of course it that it is matching the colors of the words with the notation. It is a helpful tool for people who cannot read the notation.