White gaussian noise analysis deduction
They say that $n(t)$ is white noise with a double-sided power spectral density (PSD) of $N_0/2$, i.e., the PSD is given by
$$S_n(f)=\frac{N_0}{2}\tag{1}$$
The auto-correlation function is the inverse Fourier transform of $(1)$, which is
$$R_n(\tau)=E\big\{n(t)n(t+\tau)\big\}=\mathcal{F}^{-1}\big\{S_n(f)\big\}=\frac{N_0}{2}\delta(\tau)\tag{2}$$