The Fourier transform maps the function s(t) in the time domain to a function S(f) in the frequency domain. We often call the function s(t) a 'waveform' and usually is some periodic function representing a wave (sound, radio, etc). What this means is that the function s(t) is broken down into a number (infinite possibly) of separate simple waves which have a single frequency. Physically, we can think of a sound from a musical instrument consistings of a base note or frequency with a number of harmonics, so we have a 'superpositioning' of a finite number of frequencies, perhaps all with different amplitudes which combine into a distinct 'tone' .
To extract the individual frequencies, we do a kind of 'correlation' of the original waveform with the simplest wave components, that is simple sine/cosine waves at each frequency - this is what is being expressed in the Fourier transform integral above. This is expressed by the term exp-i2*pi*f
which is the complex representation of sin and cos. So, by integrating the waveform with the exponential, this gives us the component of the waveform for a particular frequency 'f'.
For a waveform based upon a single sine wave at frequency 'F', you should be able to see that the result of the integration will be a function zero everywhere except for F. As the operation is linear, if the waveform is a linear superposition of multiple sin waves of different frequencies, the result of integration will be a function which is zero everywhere except for the points corresponding to the individual frequencies. Hence the transform function S(f) represents the distribution of frequencies of the underlying waveform.