Reconstruction of Contiuous - Time Signals
Because sampled signals Spectrum will have copies of original spectrum at multiples of $f=F_s$. Even if aliasing happened due to choice of $F_s$, the spectrum of sampled signal $x(n.T_s)$ will be periodic in frequency always with period $F_s$. And you can take any one such $k^{th}$ copy or period and integrate from $\frac{k-1}{2}F_s$ to $\frac{k+1}{2}F_s$. Now if you vary $k$ from $-\infty$ to $\infty$, then it is equivalent to integration on LHS. 

Key: Sampling at $F_s$ will create a spectrum which is periodic in frequency with period $F_s$.