I'm trying to better frame the Fourier Transform in light of representation theory. The classical definition that I remember from the course I had attended many years ago is
\begin{align}\mathcal{F}: L^1(\mathbb{R}) &\rightarrow \mathcal{C}_0(\mathbb{R}) \\ f&\mapsto \int_{\mathbb{R}}f(x)e^{-2\pi i\xi x} \, dx\end{align}
where $\mathcal{C}_0(\mathbb{R})$ denotes the space of continuous functions which approach $0$ at $\pm \infty$.
Now, by reading here and there, I kinda got the intuition that $e^{-2\pi i \xi x}$ are the irreducible representations of the (locally-compact) group $(\mathbb{R},+)$ (not sure about it).
Why do we use them to define the Fourier transform?
Can anybody help me to fill this gap?