What does the exponential term in the Fourier transform mean?
It's a complex exponential that rotates forever on the complex plane unit circle:
$$e^{-j\omega t} = \cos(\omega t) - j \sin(\omega t).$$
You can think of Fourier transform as calculating correlation between $f(t)$ and a complex exponential of each frequency, comparing how similar they are. Complex exponentials like that have the nice quality that they can be time-shifted by multiplying them with a complex number of unit magnitude (a constant complex exponential). If the Fourier transform result at a particular frequency is a non-real complex number, then the complex exponential of that frequency can be multiplied by that complex number to get it shifted in time so that the correlation to $f(t)$ is maximized.