Why does waverly get upset and run away from her mother in the marketplace

The most direct reason of why Waverly ran away was because she was embarrassed and annoyed at her mother.  Waverly did not like her mother boasting about her and her chess prowess.  She wanted to be her own person. We can see this as Waverly says these words to her mother in exasperation:

"Why do you have to use me to show off? If you want to show off, then why don't you learn to play chess?"

Even though this was what made Waverly run away, the truth of the matter is that Waverly was getting annoyed for a while.  Her mother was too overbearing in her mind. For example, according to Waverly, Mrs. Jong had a habit of standing over her while she was practicing chess.  Her breathing annoyed her as well as her sounds.  This shows that Waverly is getting bitter. When she could not take it any more, she confronted her mother and ran away.My mother had a habit of standing over me while I plotted out my games. I think she thought of herself as my protective ally. Her lips would be sealed tight, and after each move I made, a soft "Hph" would escape from her nose.With this said, Waverly also knew deep in her heart that she made a mistake. She said:I knew it was a mistake to say anything more, but I heard my voice speaking...

In light of this, Waverly's embarrassment of exploding might have caused her to run away as well. 

 source for info: enotes.com