In this problem, a subgraph of $G = (V,E)$ is given by $G' = (V', E')$ where $V' \subset V$ and $E'$ is subset of edges of $E$ that connect two vertices in $V'$. How many subgraphs of $K_{m,n}$ are there that contain $m+n$ vertices?
This document (problem 30) gives the answer $2^{mn}$. I understand that that the number of edges in the graph is $mn$, but I don't understand how $2^{mn}$ counts only the subgraphs where all $m+n$ vertices are included. Is a single-line subgraph included in $2^{mn}$ subgraphs? That would only have 2 vertices instead of $m+n$ vertices.
You know already that you have $m+n$ vertices and $mn$ edges in your graph. If you want your subgraph to contain $m+n$ vertices, then $G'$ will contain all vertices of $G$. Because of this every edge of $G$ can be in $G'$ (if we delete one vertex, the edges incident with this vertex can not be in $G'$).
So for every edge $e$ of $G$ we have two possibilities: either $e$ is in $G'$ or not. Thus there are
$$ \prod\limits_{e\in E} 2 = 2^{|E|} = 2^{mn} $$
possible ways of forming such a subgraph.