Assume you model the likelihood through the sufficient statistic $\bar{X}$:
\begin{equation}
P(D|\mu) \propto \textrm{Exp}(-\frac{n}{2\sigma²}(\overline{x}-\mu)^2) \propto N(\overline{x},\theta,\frac{1}{n})
\end{equation}
And use the following as a prior for $\mu$:
\begin{equation}
P(\mu) \propto \textrm{Exp}(-\frac{n}{2\sigma_0^2}(\mu-\mu_0)^2) \propto N(\mu|\mu_0,\sigma_0^2)
\end{equation}
We need to show that the product of these two distributions has a kernel which is the kernel (as a function of $\mu$) of a normal distribution; so we can ignore all of the normalizing constants and any terms not containing $\mu$.
\begin{equation}
\begin{split}
P(D|\mu)P(\mu) & \propto \textrm{Exp}(-\frac{n}{2\sigma²}(\overline{x}-\mu)^2) \textrm{Exp}(-\frac{n}{2\sigma_0^2}(\mu-\mu_0)^2) \\
& = \textrm{Exp}(-\frac{n}{2}[\frac{1}{\sigma^2}(\bar{X}^2 - 2\bar{X}\mu + \mu^2)+\frac{1}{\sigma_0^2}(\mu^2 - 2\mu\mu_0 + \mu_0^2)])
\end{split}
\end{equation}
Throwing away terms which do not contain $\mu$, we get:
\begin{equation}
\begin{split}
\textrm{Exp}(-\frac{n}{2}[\frac{1}{\sigma^2}(2\bar{X}\mu + \mu^2)+\frac{1}{\sigma_0^2}(\mu^2 - 2\mu\mu_0)]) & \propto \textrm{Exp}(-\frac{n}{2}[\mu^2(\frac{1}{\sigma^2} + \frac{1}{\sigma_0^2}) +\mu(\frac{2\bar{X}}{\sigma^2}-\frac{2\mu_0}{\sigma_0^2})])
\end{split}
\end{equation}
Now to get a quadratic in $\mu$ (and thus an appropriately specified normal distribution) we complete the square:
\begin{equation}
\begin{split}
\textrm{Exp}(-\frac{n}{2}[\mu^2(\frac{1}{\sigma^2} + \frac{1}{\sigma_0^2}) +\mu(\frac{2\bar{X}}{\sigma^2}-\frac{2\mu_0}{\sigma_0^2})]) & = \textrm{Exp}(-\frac{n}{2}[a(\mu-h)^2 + k])
\end{split}
\end{equation}
For some $a$, $h$, and a term $k$ which does not involve $\mu$ and can be removed.
Specifically, ignoring $k$, we have that $a = (\frac{1}{\sigma^2} + \frac{1}{\sigma_0^2})$, and that $h = -\frac{(\frac{2\bar{X}}{\sigma^2}-\frac{2\mu_0}{\sigma_0^2})}{2(\frac{1}{\sigma^2} + \frac{1}{\sigma_0^2})}$.
This gives us our mean ($h$), and the reciprocal of our variance (up to rescaling by the $n$ factor present):
\begin{equation}
\begin{split}
\textrm{Exp}(-\frac{n}{2}[\mu^2(\frac{1}{\sigma^2} + \frac{1}{\sigma_0^2}) +\mu(\frac{2\bar{X}}{\sigma^2}-\frac{2\mu_0}{\sigma_0^2})]) & \propto \textrm{Exp}(-\frac{n(\frac{1}{\sigma^2} + \frac{1}{\sigma_0^2})}{2}[(\mu-\frac{(\frac{2\bar{X}}{\sigma^2}-\frac{2\mu_0}{\sigma_0^2})}{2(\frac{1}{\sigma^2} + \frac{1}{\sigma_0^2})})^2])
\end{split}
\end{equation}
Which is a $\textrm{N}(\frac{(\frac{2\bar{X}}{\sigma^2}-\frac{2\mu_0}{\sigma_0^2})}{2(\frac{1}{\sigma^2} + \frac{1}{\sigma_0^2})}, (\frac{1}{n}(\frac{1}{\sigma^2} + \frac{1}{\sigma_0^2})^{-1})$.
Which corresponds to the result given here up to some canceling of values. The conjugate prior specified in the problem also has the $n$ factor, which the Wikipedia page doesn't have. You can just collapse that into $\sigma_0$, though, to get the same result they have.