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would someone be so kind to explain this to me: enter image description here

Especially the arg min part.

(it's from the k-means algorithm)

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up vote 5 down vote accepted

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 it's 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 is helped.

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