Waverly starts to win more chess games because of three reasons. 

First, Waverly was eager to learn; she was an apt pupil.  She learned well from Lau Po.  Po was a decent player, and she soon exhausted his knowledge.  Here is what the text says:

By the end of the summer, Lau Po had taught me all he knew, and I had become a better chess player.

Second, Waverly learned the art of invisible strength from her mother as a little girl, and more importantly she applied this knowledge to chess to her advantage.  Because of this, she became very clever. Waverly admits this right from the beginning of the story. She says:

I was six when my mother taught me the art of invisible strength. It was a strategy for winning arguments, respect from others, and eventually, though neither of us knew it at the time, chess games.

Finally, Waverly was obsessed with chess. This is probably the greatest reason why she began to win more.  In one section of the story, she says that she drew a chessboard and pinned it next to her bed.  From this board she would think about chess and create mock battles in her mind. Again she admits:

I loved the secrets I found within the sixty-four black and white squares. I carefully drew a handmade chessboard and pinned it to the wall next to my bed, where I would stare for hours at imaginary battles.