This is a special case of Ryser's Conjecture, which states that in an $r$-partite, $r$-uniform hypergraph (with $r>1$)
$\tau \leq (r-1) \nu$,
where $\tau$ is the size of a minimum cover and $\nu$ is the size of a maximum matching. Note that the case $r=2$ is simply König's theorem.
You are interested in the case $r=3$, which was settled by Aharoni. The reference is
Ryser's conjecture for tripartite 3-graphs. Combinatorica 21 (2001), no. 1, 1--4,
and you can find a copy of the paper here.