This is colloquial speech. This is the way how people think and speak in real life. When you talk to a friend, is it very unlikely, that you always produce grammatically correct and complete sentences.
The author wanted to write the story in the most involving way. She wanted to make the reader feel what the woman felt. So, the author used two methods to let the reader dive really deep into the world of that woman:
first-person narrator
The hero of the story and the narrator are the same person. Reading such a text is like listening to someone who tells a real story about herself (or himself).
colloquial speech
Normally when you write down something, you write down filtered speech. Spoken language and written language are very different. Written language is much more strict and obeys grammatical rules much more than spoken language. In spoken language you use less full sentences but more fragments of sentences. And not every sentence you speak is perfect. People who produce grammatically perfect sentences when they speak are rare (Klaus Albrecht Schröder, director of Albertina, a famous museum in Vienna, is such a person. Listening to him feels strange.)
So, when the author doesn't write grammatically perfect sentences, but writes more the way how people speak, this feels much more natural. It feels more like if the woman really tells her story. The reader thus experiences this woman more directly and is also emotionally closer to her as a result.
Analysis of the concrete sentence:
Ich rieb, bis mir der Arm schmerzte.
I rubbed until my arm hurt.
This is a normal and absolutely correct sentence. The part »der Arm« is a nominal group in nominative case, and it is the subject of the subordinate clause. But we can replace this nominal group with an enumeration. Since the subject now contains more than only one thing, the verb must turn from singular to plural (schmerzte → schmerzten).
Ich rieb, bis mir der Arm und meine Finger schmerzten.
I rubbed until my arm and my fingers hurt.
But what, if you tell about your arm hurting, and at the end of the sentence it comes to your mind, that your fingers hurt too? The verb in singular has already left your mouth, and you want to tell about the fingers too. This is a situation that you will experience many times when you speak free to somebody. A grammatically correct solution would be this:
Ich rieb, bis mir der Arm schmerzte und bis mir meine Finger schmerzten.
I rubbed until my arm hurt and until my fingers hurt.
But this sounds unnatural and it is too long for just a little add-on to the main idea that already has been said. The words bis, mir and schmerzte(n) have already been said, so what you do in colloquial speech is just not repeat them, you omit their second appearance, and what you get is this:
Ich rieb, bis mir der Arm schmerzte und meine Finger.
I rubbed until my arm hurt and my fingers.
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und meine Finger." Then it would make sense to add some kinda intensification, if not, this isn't a grammatically correct sentence in German ;). Normally as you mentioned it should be: Ich rieb, bis mir der Arm und meine Finger schmerzten
.