I am reading "introduction to graph theory" written by Douglas. I try to understand the proof of 1.3.19 theorem from this book but I failed... Here is the statement.

$\text{Every loopless graph G has a bipartite subgraph with at least e(G)/2 edges.}$

Author starts with any partition $V(G)$ into two sets $X,Y$. Using the edges having one endpoint in each set yields a bipartite subgraph $H$ with bipartition $X,Y$. He said that if $H$ contains fewer than half the edges of $G$ incident to a vertex $v$, then $v$ has more edges to vertices in its own class than in the other class. Moving $v$ to the other class gains more edges of $G$ than it loses.

I do not understand why the last two sentences are true. Could anyone give some example or detailed description about this theorem? Thanks!