In principle, understanding Fourier transform doesn't require understanding distributions and functionals, it's understanding of distributions that requires understanding of functionals, and the space of distributions $S'$ is a convenient place to study Fourier transform because it is invariant under it. It's the Fourier transform itself that relates to frequency and time, informally it decomposes "signals" in time into single frequency "waves", so you can think of argument of the original function as time, and of the Fourier transform's as frequency.
Another space invariant under Fourier transform is $L^2$, Fourier transform even preserves the $L^2$ norm and you can understand it there without distributions and functionals. But it is much smaller than $S'$, in particular it does not contain Dirac's delta functions or pure harmonics $e^{i\omega t}$. Interestingly enough on $S'$ Fourier transform maps them into each other, which makes sense since frequency of a pure harmonic should be concentrated at a single point. Distributions like delta functions are most often interpreted as functionals over spaces of "nice" test functions like $S$, it makes it easier to transfer constructions and properties from test functions to distributions. For example, that $S'$ is invariant under Fourier transform immediately follows from the fact that $S$ is. So it pays off to understand how functionals work, perhaps with a different book.
