Interpreting the amplitude of signals in fourier transform
"I totally understand the concept of Fourier transform"


Lucky you if you really do. Some of us (me, in first place) don't (in totality). The Fourier transform (and its avatars) is a prototype for duality. Duality here means that you can represent a signal on some primal domain (time) onto a dual domain (here frequency). This transformation is meant to possess useful properties, to preserve signal information, and to add insight, like a somewhat easier interpretation.

Signals are traditionally represented as "amplitudes of something for each time". These amplitudes are not always absolute. But we can hope they are linearly related to the actual physical values. And that the linearity factor does not change over time. Under the above premises, sines and cosines, or complex exponentials (cisoids), are a very good way (perhaps the best) to model  the system. And in turn they can be used to represent the data, as a linear combinaison of shifted sines.

Since the sines are orthogonal, the squared magnitude of the coefficients weighting the sines is proportional to the energy of each corresponding sine frequency.

The proportional factor depends on how the coefficients are computed, often up to a scaling factor. The amplitude of a given frequency component can be directly computed with correlation. Note however that this coefficient should be taken with care, as outer conditions play a role: the finite horizon observation of a function/signal, amplitude discretization, noise, etc.