I've been self-studying Fourier Transform for a while ("vanilla" transform, not FFT, DFT or other types of that sort) but I cannot seem to understand how the mathematical expression reflects the idea. Most sources that claim to be "intuitive" seem to recite similar ideas, like wrapping the function around a circle or representing functions as movements in circular patterns, etc. I kind of understand that process, but I cannot understand how it is reflected in the equation. I cannot understand the phrase " $$(e^{-2\pi(i)\alpha x})$$ represents wrapping the input wave function $f(x)$ around the origin of the complex plane at some frequency $\alpha$ " which has been shown here, in particular. I know that at the heart of it lies the Euler's formula, which represents rotations with a given frequency.
How does it imply "wrapping"? How does this relate to the rest of the formula? I think my confusion is caused by notation, rather than the concept itself, but please feel free to correct me.
If it comes to visualization $3$Blue$1$Brown is one of the best. Here he talks about the Fourier Transform and explains the idea of wrapping intuitively (Minute: 2:55). Also this video may give you a new perspective too.