In addition to the other content-ful answers and remarks, especially to amplify @ToddTrimble's reminder about J. Tate's thesis: it is no secret that A. Weil, R. Godement, E. Artin, and K. Iwasawa had been thinking about representation theory of locally compact groups for at least a decade before Tate's thesis. E. Artin's student Margaret Matchett's thesis at Indiana Univ. in 1946 already used similar ideas, and K. Iwasawa's ICM talk in 1950 exactly referred to such an approach to zeta functions. (Thus, for quite a few years now, I've suspected that Tate's disinterest in publication of his thesis was his awareness of the "prior art" that made the thesis less novel... and he had many other things in his mind.)
Already in the early 19th century, and perhaps in the late 18th (Euler et al), there was a visible intuitive appreciation of the nearly-magical conversion by Fourier transform of smooth-to-decay properties. In particular, cancellation properties were implicit and understood.
Wiener's and Bochner's careful founding of Fourier transform theory, and then Schwartz', made the subtle features of van der Corput's (and Landau's and Hardy-Littlewood's earlier) expeditions more persuasive.
(I am not competent to talk about the details of the history of "the circle method", although I am aware that this has played a significant role, with significant recent advances...)
Hecke's, Siegel's, and Maass' collective work from c. 1920 through 1960 arguably amounted to applications of harmonic analysis in a broad sense (as opposed to really algebraic forms of algebraic geometry).
Selberg's and Roelcke's ideas about harmonic analysis of automorphic forms in the late 1950s, stimulated by Maass' ideas to a considerable degree, did not quite succeed in bearing on classical issues such as RH, but gave many provocative insights.
R. Langlands' fundamental ideas in the mid-1960s in many ways were a fulfillment and precisification of general pronouncements of Selberg, with quite a few surprises. Harish-Chandra appreciated and documented and amplified this. Both parties had prior experience with issues of representation theory of real Lie groups, with nascent appreciation for representation theory of p-adic groups.
Duh, much more can be said, but the point is that it is 150+ years of experience that shows that "Fourier analysis" or other modern extrapolations of it are not merely "helpful", but essential to understanding.
A philosophical explanation? I don't know. To say that various conjectures of Langlands or others are adequate to explain the relevance I think is significantly inadequate... though not counter-factual.
In my own little bailiwick, the idea that taking a special function on a bigger space (e.g., Siegel-type Eisenstein series on symplectic group) and restricting it to smaller reasonable space (product of smaller symplectic groups) and decomposing it into eigenfunctions (for various operators), turns out to present some L-functions as decomposition coefficients.
That is, who knew?, but L-functions and zeta functions turn out to be slightly-more-approachable by observing them as decomposition coefficients...
Such stuff... :)