Why does Betsy leave the mill in Lyddie?

Betsy leaves the mill because she is too sick to work and wants to go to college.

Betsy is a factory worker and a friend of Lyddie’s.  She is interested in Lyddie right away.  Lyddie shares a bed with Betsy.  She likes to read novels, and teaches Lyddie how to read better.  Lyddie really enjoys Oliver Twist because she relates to the character.

Betsy admits to Lyddie that she is interested in going to college.

...

Betsy leaves the mill because she is too sick to work and wants to go to college.

Betsy is a factory worker and a friend of Lyddie’s.  She is interested in Lyddie right away.  Lyddie shares a bed with Betsy.  She likes to read novels, and teaches Lyddie how to read better.  Lyddie really enjoys Oliver Twist because she relates to the character.

Betsy admits to Lyddie that she is interested in going to college.

She, too, was saving her money, she confessed quietly to Lyddie and asked her not to tell, to go for an education. There was a college out West in Ohio that took female students‐a real college, not a young ladies' seminary. (Ch. 11)

Betsy tells them one day that she is worn out. She is interested in signing the petition because she is tired of longer and longer work days.  She is more interested in going to college.  As soon as she saves up her money, she plans to go.  She has been putting her brother through Harvard already.

"If they dismiss me, I'd have to stop stalling and blathering and get myself to Oberlin College and a new life." By now, Lyddie was propped up on her elbow listening, torn between pride for Betsy and horror at what she was proposing. "So, you're awake after all, our sleeping beauty." (Ch. 13)

Betsy has a bad cough.  She has nothing to lose, so she signs the petition.  This means that she is dismissed and cannot work in the factory again, but it is too hard for her anyway.  She has to leave the mill for good.  She does not like the hospital because it is too expensive.

Betsy may never have gotten to college.  She could have died from the illness, but either way she did not have enough money to go to college at the time she quit and being sick cost her money.