What does $|V|=O(|E|)$ mean?
First, $|V|$ is the number of vertices and $|E|$ is the number of edges.

The point is that if a graph is connected it must have at least $|V|-1$ edges.  Therefore, $|V|\leq |E|+1$, so
$$|V|\log|V| + |E|\log|V| \leq 2(|E|+1)\log|V|\leq 3|E|\log |V|\,.$$

Writing $|V|=O(|E|)$ is something of an abuse of notation. $O(\cdot)$ is an asymptotic statement about what happens as some variable becomes large, over some infinite family of instances.  So, here, we're supposed to imagine that the infinite family of instances is the set of all connected finite graphs. It would have been clearer to just write $|V|\leq |E|+1$.