The power source for the mills is the running water from the river. When Mrs. Bedlow, the boarding housekeeper, asks Lyddie for a hand, she explains why:
I think the girls will be home for breakfast early. The river's too high and the mill wheels are likely slowing.
Here's a short explanation (with images) of how water mills worked in the 19th Century. During this time period, the development of new techniques to harness water power for running textile mills allowed the Massachusetts Waltham-Lowell System to compete with the British textile industry.
As in the story, the Lawrence and Lowell textile mills depended on the labor of young women. These young women lived in boarding houses such as the one Lyddie finds herself in. Boarding house matrons such as Mrs. Bedlow make sure that company curfews and expectations are fulfilled to the letter. For example, when Lyddie admits that she has not made plans to attend church, Betsy warns her that
...regardless of the state of your immortal soul, the corporation requires regular attendance of all its girls.
For more on water-powered mills, please refer to the link below.