a great answer to this question probably does not yet exist because its a relatively young and very active area of research. for example Ingo Wegeners comprehensive book on boolean functions from 1987 has nothing on the subject (except for analyzing the circuit complexity of the DFT).
a simple intuition or conjecture is that it appears that large Fourier coefficients of higher order indicate the presence of subfunctions that must take into account many input variables and therefore require many gates. ie the Fourier expansion is apparently a natural way to quantitatively measure the hardness of a boolean function. have not seen this directly proven but think its hinted in many results. eg Khrapchenkos lower bound can be related to Fourier coefficients.
another rough analogy can be borrowed from EE or other engineering fields to some degree where Fourier analysis is used extensively. it is often used for EE filters/signal processing. the Fourier coefficients represent a particular "band" of the filter. the story there is also that "noise" seems to manifest in particular ranges of frequencies, eg low or high. in CS an analogy to "noise" is "randomness" but also its clear from much research (reaching a milestone in eg ) that randomness is basically the same as complexity. (in some cases "entropy" also shows up in the same context.) Fourier analysis seems to be suited to study "noise" even in CS settings.
another intuition or picture comes from voting/choice theory.[2,3] it is helpful to analyze boolean functions as having subcomponents that "vote" and influence the outcome. ie analysis of voting is a sort of decomposition system for functions. this also leverages some voting theory which reached heights of mathematical analysis and which apparently predates the use of much Fourier analysis of boolean functions.
also, the concept of symmetry appears to be paramount in Fourier analysis. the more "symmetric" the function, the more that Fourier coefficient cancel out, and also the more "simple" the function is to compute. but also the more "random" and therefore more complex the function, the less the coefficients cancel out. in other words symmetry and simplicity, and conversely asymmetry and complexity in the function seem to be coordinated in a way that Fourier analysis can measure.
 On the Fourier analysis of boolean functions by Bernasconi, Codenotti, Simon
 A brief introduction to Fourier analysis on the Boolean cube (2008) by De Wolf
 Some topics on the analysis of boolean functions by O'Donnell
 Natural proofs by Razborov & Rudich