Explanation on arg min

$\arg\min$ is argument of the minimum.

The simplest example is

$\arg\min _{x} f(x)$ is the value of $x$ for which $f(x)$ attains its minimum.

for your example

$x_n$ is known and depends on $\pi_{nk}$
and $k$ equals to $j$ such that $\begin{Vmatrix} x_n-\mu_j \end{Vmatrix}^2$ attains minimum among all values of $\mu_j$ and given $x_n$.

hopefully that helps.