Fourier transform of the sampled signal
The equality
$$\frac{1}{T}\sum_{k = -\infty}^{\infty}X(j(\omega - k\frac{2\pi}{T})) = \sum_{k = -\infty}^{\infty}x(kT)e^{-j\omega kT}\tag{1}$$
is an instance of Poisson's sum formula. The term on the right-hand side of $(1)$ is just the Fourier series representation of the periodic function on the left-hand side of $(1)$.
The samples $x(kT)$ of the time domain signal are basically the Fourier series coefficients of the periodic spectrum of the sampled signal.
The significance of $(1)$ is that it shows that the discrete-time Fourier transform (DTFT) $X_d(e^{j\Omega})$ of the sequence $x_d[k]=x(kT)$ equals a periodized version of the Fourier transform of the corresponding continuous-time signal $x(t)$:
$$X_d(e^{j\Omega})=\sum_{k=-\infty}^{\infty}x_d[k]e^{-jk\Omega}=\frac{1}{T}\sum_{k=-\infty}^{\infty}X\left(\frac{j(\Omega-2\pi k)}{T}\right),\qquad\Omega=\omega T\tag{2}$$
Clearly, if $X(j\omega)$ is band-limited with a maximum frequency $\omega_c&lt;\pi/T$, then the shifted spectra in the sum on the right-hand side of $(2)$ won't overlap, i.e., there is no aliasing and the signal $x(t)$ can be reconstructed without error from its samples $x(kT)$. That means that the basic form of the sampling theorem is implicit in Eq. $(2)$.