From here on, $n$ will denote the number of vertices and $k$ will denote the number of connected components of the graph in question.
Let $F$ be a forest, then $F$ has $n-k$ edges.
Every connected component of $F$ is a tree, and as trees are planar, $F$ is planar. Then we can apply Euler's formula: $n-e+f=1+k$.
As $F$ has no cycles, we have that $f=1$ and thus $e=n-k$.
Until here I understand everything, but the following in troubling me.
It follows that every simple graph has at least $n-k$ vertices.
How does that follow from the theorem above?