Try proving that the graph can be drawn by induction: given a suitably chosen drawing of the $n$-vertex $G$, it can be extended to the $(n+1)$-vertex $G$ by adding a vertex $v_{n+1}$ adjacent to $v_{n-2}, v_{n-1}, v_n$.
For this to be possible, $v_{n-2}, v_{n-1}, v_n$ all have to lie on the same face of the $n$-vertex graph. Actually, since they are all adjacent, and all faces of $G$ are triangles, they must be a face of the $n$-vertex graph.
For simplicity, we may assume that this face is the external face, which makes adding $v_{n+1}$ easy. So we have the strengthened claim:
Claim. For all $n\ge3$, $G$ has a plane embedding in which the external face is a triangle with vertices $v_{n-2}, v_{n-1}, v_n$.
Try to prove this claim by induction.