To add a representation theory perspective: if $G$ is a Lie group, and $f$ is a function (or more precisely a distribution) on $G$ then (under certain mild conditions on $f$ and $G$), the function $f$ is uniquely determined by its unitary matrix coefficients, i.e. the coefficients of the matrix $\rho(f)$ where $\rho:G\to GL_n$ goes over all isomorphism classes of unitary and irreducible representations. This perspective should be understood as a change of basis that reveals the "underlying equivariant properties" of a function $f$, i.e. the properties important from the point of view of representation theory.
Now the unitary irreducible representations of the additive group $\mathbb{R}$ are one-dimensional representations $\rho_\alpha: t\mapsto e^{i\alpha t}$ indexed by $\alpha\in \mathbb{R}$, and so the matrix coefficient decomposition of a function is precisely its Fourier transform. This hints that whenever you are interested in problems with additive equivariance (action by $\mathbb{R}$), you should expect to see Fourier transforms.