Here is a crude way to see that this is true: Given $a\in A$ and $b\in B$, consider $A-a =\{x-a | x\in A\}$ and $b-B =\{b-y | y\in B\}$. These shifted triangles have a convex intersection that is more than a point because $2+2 \gt 3$. The intersection gives the pairs of points adding up to $a+b$, and each extreme point of the intersection corresponds to an extreme point of $A$ or $B$, hence a point contained in an edge of $A$ or an edge of $B$.

I think there is a nicer argument in terms of the projection of $A \times B$ to the Minkowski sum.