Free market capitalism is a system that is based on supply and demand, with the market settin the cost of different products. In this system, it is beneficial to manufacture products at as low of a cost as possible. One way to keep manufacturing prices low is to use cheap labor sources, and private prisons are one such source.
Private prisons are run by corporations as opposed to being run directly by the government. These corporations are paid by the government to manage the prison, but they can make extra profits from inmate labor. Inmates do earn wages, but they are extremely low—according to the Civil Liberties Defense Center, the average pay for an inmate in 2016 was less than one dollar per hour (link below).
The issue of labor in private prisons is relevant to sociologists not only because the corporations that run them are motivated by profit, but because there tends to be less accountability for private prisons—this means that there are discrepancies in how prisoners are treated depending on whether they are incarcerated in a public prison or a private one. Private prisons tend to have harsher working conditions for their inmates and fewer regulations for their staff, which can lead to the abuse and exploitation of prisoners. The second link below has more information on this (and on the Fair Labor Standards Act, which has been interpreted ambiguously with respect to prisons).
Overall, the controversy surrounding private prisons has long been of interest to sociologists due to the conditions experienced by prisoners and the motives and accountability (or lack thereof) of the corporations responsible for them. Since these issues are ongoing, this topic is definitely worth studying further.