Proof

By hypothesis, $G(V,E)$ is a simple, biconnected, and planar graph.
Let $\mathcal D$ be a planar drawing of $G$.
Let $|F|$ be the number of faces in the drawing $\mathcal D.$
Since a simple graph has no loops or multiple edges, each face in $\mathcal D$ has a face degree of at least $3.$
It follows from the handshaking lemma for planar graphs that $3|F|\le 2|E|.$
Using the Euler's formula for planar graphs, we have $|F|=|E|-|V|+2.$
It follows $3|E|-3|V|+6\le 2|E|,$ which is equivalent to $|E|\le 3|V|-6.$
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Aldous Joan M., Wilson Robin J.: "Graphs and Applications - An Introductory Approach", Springer, 2000