“Thumping against his side, she twisted and finally caught enough breath to begin to complain. She yelped her confusion at first, and her breath ignited, she began to wail, and to try to twist her arm free.”
In your sentence ignited basically means “started”. When you ignite a fire, you start a fire. Let’s take a look at some other words in the sentence:
Breath basically means “voice”.
First her voice was heard as a yelp (a very short sound/cry). You might compare the length of a yelp to a spark.
Then her voice became a wail (a prolonged sound/cry). You might compare the duration of wailing to the duration of fire/burning - i.e., much longer than a spark.
So, ignited refers to the start of the more prolonged sound (wailing).