Argmax in $k$-NN algorithm
Given a new point $x_q$, we find the $k$ closest points $x_1,\ldots,x_k$. Each one of them is associated with a cluster $f(x_1),\ldots,f(x_k)$. We assign $x_q$ to the most common cluster among $f(x_1),\ldots,f(x_k)$. The argmax equation is just a fancy way of expressing this idea: for any cluster $v$, the quantity $\sum_{i=1}^k \delta(v,f(x_i))$ is just the number of $x_i$'s which are assigned to cluster $v$, and the argmax picks up the cluster $v$ maximizing this parameter.