Say it is known that some state $x$ is normally distributed with mean $\mu$ and variance $\sigma^2$. Furthermore, assume that $\sigma^2$ is known, but $\mu$ is not. However, there is a prior for $\mu$, which is that $\mu \sim N(m, s^2)$.
Now, we observe $n$ draws from the distribution for $x$, where, $x \sim N(\mu, \sigma^2)$. What inference can we draw regarding the values of $\mu$?
Using Bayes' rule you can define
$$f(\mu|x,\sigma^2) = \frac{f(\mu,x|\sigma^2)}{f(x|\sigma^2)} = \frac{f(x|\mu,\sigma^2)f(\mu|\sigma^2)}{f(x|\sigma^2)}$$
Now
$f(x|\mu,\sigma^2) = N(\mu,\sigma^2)$
$f(\mu|\sigma^2) = f(\mu) = N(m,s^2)$
$f(x|\sigma^2)$ is a constant that you don't need
Therefore you can see that
$$f(\mu|x,\sigma^2) \text{ is proportional to } N(\mu,\sigma^2) \times N(m,s^2)$$
And you can see from these notes (pages 1-3) that it follows a Normal distribution.
