We'll begin by taking $G=(V,E)$ to be any sub-cubic planar graph (i.e., $\Delta(G)\leq 3$), and embed it into the plane. Call the image of this embedding $\Gamma$ (so $\Gamma$ is a collection of points and curves in the plane).
(Step 1) We're going to modify the embedding $\Gamma$ to create a `nicer' plane graph, one where the edges are all straight lines near to any vertex. This first step can be skipped if you're happy to just apply Fary's Theorem and get a straight-line embedding of $G$.
Let $\delta = \min\{||x-y||: \text{$x$ is a vertex of $\Gamma$, $y$ is on an edge not incident with $x$}\}$. In other words, $\delta$ is the minimum distance between any vertex, and a point on an edge that doesn't touch that vertex.
Around each vertex $v$ of $\Gamma$, draw a circle $S$ of radius $\frac{\delta}{2}$ centered at $v$. We're going to modify all the edges incident with $v$ as follows:
If $uv$ is an edge starting at $u$ going to $v$, consider the first point $p$ on $uv$ that touches $S$. Delete the whole segment of the edge that goes from $p$ to $uv$, and replace it with a straight line segment $[p,v]$. Do this for every edge incident with $v$, and every vertex $v$. Call the new plane graph $\Gamma'$, and note that it too is an embedding of $G$ (so it's the same graph, just drawn differently).
(Step 2) We're going to create a plane graph $\Gamma^*$ that has the line graph $L(G)$ as a minor - since every minor of a planar graph is planar, this will show that $L(G)$ is planar.
To create $\Gamma^*$ from $\Gamma'$, first put a vertex $\overline{e}$ in the middle of each edge $e$. Then, place a circle of radius $\frac{\delta}{4}$ around each vertex $u$ of $\Gamma'$. If the circle intersects the edge $e$ of $\Gamma'$ near to $u$, make this intersection point a new vertex called $u_e$. The arcs of the circle are now edges of a plane graph $\Gamma^*$ containing $\Gamma'$ as a subdivision.
To see that the line graph of $G$ is a minor of $\Gamma^*$, do the following:
Remove every vertex $u$ of the original plane graph $\Gamma'$, and all the mini-edges $\{u,u_e\}$ incident with one of the original vertices. Then, contract every edge of the form $\{\overline{e}, u_e\}$. Finally, throw out any loops or duplicate edges you might have from the circles around vertices of degree 1 or 2. What's left is a plane embedding of $L(G)$.
