We have a directed simple graph $G$. Out-degree of any vertex is at most $k$ in $G$. We need to show that $G$ is $2k+1$-partite.
Attempt it by induction. Assume the property is true for every graph with less than $n$ vertices. Now let $G$ be a graph with $n$ vertices. Take the vertex $v$, with the smallest in-degree.
How big can the in-degree of this smallest $v$ be? And you know the outdegree of $v$ is at most $k$. So how many vertices is it adjacenct to? Now $G$ minus $v$ is a graph of $n-1$ vertices and so it is coloured with your $2k+1$ colours. Now adding that $v$ back.... (you can finish from here)