Fourier series expansion of $\exp[-j2kA\sin(\omega t)]$
They're talking about the Fourier series expansion of the periodic function

$$f(\tau)=e^{-jx\sin(\tau)}=\sum_{n=-\infty}^{\infty}c_ne^{jn\tau}\tag{1}$$

which sometimes appears in the literature, e.g., when computing the spectrum of a frequency modulated sine wave. The complex Fourier coefficients in $(1)$ are given by

$$c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\tau)e^{-jn\tau}d\tau=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{j(-n\tau-x\sin(\tau))}d\tau=J_{-n}(x)\tag{2}$$

where the last equality follows from the integral representation of the Bessel function of the first kind of order $n$.

Combining $(1)$ and $(2)$ we get

$$f(\tau)=\sum_{n=-\infty}^{\infty}J_{-n}(x)e^{jn\tau}\tag{3}$$

With $x=2k_0A$ and $\tau=\omega_Dt$ you obtain the expansion used in the paper you quoted.